In this screencast we'll take a look at how modifying the characteristic equation can
have an impact on the stability limits of a controlled process. So in a previous screencast
we did an analysis of a temperature control loop which showed that the ultimate gain and
ultimate period were eighty three point two percent controller output per percent transmitter
output for the ultimate gain, and eighteen point two seconds for the ultimate period.
So in that problem we developed the transfer functions provided on the screen for the process,
the valve, and the transmitter, and we're making a change to our process that we're
getting a transmitter that has less of a time constant. In other words it reacts a bit quicker.
So what's happening here is we have a new sensor/transmitter with the same range, but with a time
constant now of one second. So if we look at our transfer functions this change in the transmitter has
no effect of the process or the valve. The only difference is that our transmitter transfer
function now has a time constant of one. So what we're going to do here is we're going
to define the ultimate gain and the ultimate period. This requires the use of direct substitution
as shown in the previous screencast. So therefore we're going to need our characteristic equation
which is one plus the open loop gain, which is one plus the transfer functions for the
process, the valve, the transmitter, and the controller, and we set that equal to zero.
So therefore substituting values in here. So as we typically do with these characteristic
equations we need to find a least common denominator which will then go away once you move it to
the right hand side and multiply by zero. So if we factor the first two terms. So when
we finish out the factorization. So what we'll now do in order to do our analysis is in order
to use direct substitution we let s equal i times omega, recalling that i squared equals
negative one. So from here we'll break this in to the real and imaginary part. Since both
parts will have to equal zero, because the total is zero, so therefore it has to be zero
in both the real and in the imaginary. Since we only have one variable in the imaginary
it can be easily solved for, in this case omega. So we'll have fifty i omega cubed equals sixteen
i omega. One of the omegas cancels as well as the i's, so that leaves us with omega squared
equals sixteen over fifty, or that omega equals the square root of sixteen over fifty, which
is approximately point five six six. We can substitute that term for omega squared into
the real equation, and this gives us that our ultimate gain is a hundred and fifty eight
point four percent controller output per percent transmitter output. To find the ultimate period
we take advantage of the relationship of two pi over the ultimate frequency. This will
become two pi over the square root of sixteen over fifty, and this will give us an ultimate
period of eleven point one seconds. So what we can now do is make a comparison between
the two cases. So look at the difference between the old transmitter and the new transmitter
with respect to the ultimate gain and the ultimate period. So with a current transmitter
we see the fact that the ultimate gain is a hundred and fifty eight point four, and
that the ultimate period is eleven point one, and recall from the original problem statement
our original ultimate gain was eighty three point two, and the ultimate period was about
eighteen point two. So what we notice here is that an increase in the ultimate gain as we
went from the old transmitter to the new transmitter, which would indicate an increase in process
stability. Why? Because there is now a much larger range over which the process can be
run without any issues of stability. So from the controller standpoint the old transmitter
we get from zero to eighty three point two, whereas now we have zero to a hundred and
fifty eight point four. So therefore this causes an increase in process stability. So in addition
to speeding up our response in this case we actually also get an increase in process stability.
This won't always necessarily be the case, it will depend on the various transfer functions.
For example, in this case here, the slowest part of the controlled system was the process,
the heat exchanger transfer function. So a change in that would have a pretty large effect
on how this process operates. So the important take home point to get from this is not only
the fact that stability changes, but it's also the fact that making any small change, here
just making a quick change in a sensor transmitter, changes the dynamics of how your process acts,
which will in turn lead to a difference in how you tune your process. So in this screencast
we showed how, by making a change in a sensor/ transmitter, it has a strong influence on the
stability of a process.