|
Let us now discuss
in more detail, the properties of the dead-beat
and Dahlin controllers, and in general,
algorithms obtained from the Synthesis
Equation.
Dead-time
compensation
As would have
been noticed, both the dead-beat and Dahlin controllers provide
dead-time compensation inherently. This is because, in the specification
of the desired responses, the dead-time is specified.
Robustness
Intuitively,
because the dead-beat controller
exacts such close control, it would also be more sensitive to
process model mismatch. The most crucial one is an error in
the time-delay between the model used in controller design and
the process. Suppose we have misjudged the delay in the process
( ),
and instead of it being 4, it is either 3 or 5.
The figures below
shows the responses obtained by the dead-beat for each of these
cases:
Dead-beat controlled response
when q=3
instead of 4
Dead-beat controlled response
when q=5
instead of 4
It can be observed
that the mismatch in delays has caused significant degradation
of control performance.
The corresponding
responses obtained using the Dahlin controller
are shown below.
Dahlin controlled response when
q=3
instead of 4
Dahlin controlled response when
q=5
instead of 4
These plots show
that the Dahlin controller is hardly affected by the mismatches
in the time-delay. Thus the Dahlin controller can be said to
be more robust.
Non
minimum Phase Systems
The Synthesis
Equation is a model based controller design technique. It
is evident from the equation:
that the design
uses the inverse of the process model. This
is essentially how it enables the user to specify the desired
closed loop response. That is, it uses the inverse model to
cancel out plant dynamics, replacing it with other characteristics
so that the desired closed loop response could be achieved.
This is in fact a very common feature in modern controller design
methodologies.
However, the
use of the model inverse poses a problem when the process zeros
lie outside the unit circle, i.e. the process is a non-minimum
phase system (one that exhibits inverse responses). For example,
a process model has the general form:
where B(z)
and A(z) are polynomials in z-1, and
k is the system delay. If the roots of B(z) lie
outside the unit circle, then when 
is inverted, the inverse
will contain
poles that are outside the unit circle. That is the controller
is unstable! Thus the Synthesis
Equation should not be used to design controllers for non-minimum
phased systems.
Effects
of sampling on Discrete Zeros
This leads to
another point about the choice of sampling intervals. It can
be shown that as the sampling interval decreases, then
a minimum phased continuous system could become non-minimum
phased when discretised. This usually happens when
the continuous system has a pole-zero excess greater than 2.
Thus, the sampling interval should be chosen with care.
Now let's take
a look at a another phenomenon that is often observed when implementing
digital controllers.
|
