In this screencast we will take a look at using controller synthesis for processes where
at first glance the control that is developed is physically unrealizable. So here what we're
going to do is we are going to apply the methodology of controller synthesis to a system where
g represents a first-order plus dead time transfer function. Recall the fact that for a first
order plus dead time transfer function, or FOPTD, the transfer function is k e to the
negative theta s divided tau s plus one. So the methodology of the controller synthesis
is we pick two things. We pick a process transfer function, and note that this process transfer
function also includes the valve and the sensor transmitter, and we pick a desired response,
and generally our default response is one over tau(c) s plus one, and then from those
two values we can get the transfer function for a controller in the PID family, which
includes the value of the tuning parameters. So the formula by which we can develop our
controller transfer function is one over g multiplied by y over x divided by one minus
y over x, where y over x is this response of one over tau(c) s plus one. So when we
do the math here we end up with one over k e to the negative theta s over tau s plus
one multiplied by one over tau(c) s plus one, divided by one minus one over tau(c) s plus
one. We could go through some algebra here to kind of get the final answer, but we're
going to stop and take a look at this first term. So if we take a look at this first term,
and we rearrange it, the first rearrangement will be tau s plus one divided by k e to the
negative theta s. If we move the exponential up what we'll end up with is tau s plus one
times e to the theta s over k multiplied by the second term which, important part here,
does not contain an exponential, so the second term is not a function of e to the negative
theta s. So what that means is that the controller would have a negative dead time, because remember
that the function form of dead time is e to the negative theta s. It is physically impossible
for this to happen, you cannot have a negative dead time, therefore this means that your
controller is impossible. We now have a physically unrealizable controller. What can we do about
it? So remember that this controller synthesis takes in two things. It takes in the g, the
transfer function, we have really not much control over this. This is how the process
acts. The second is the response, and this we do have some play with because the fact that
we picked it. So recall that we set this transfer function as one over tau(c) s plus one. However
if you think about this transfer function for a second what you realize is that picking
a transfer function like this makes no sense because our process has dead time. So if our
process has dead time than it makes no sense for our response to occur instantaneously.
So what we're going to do is change our response so it has dead time, and the simplest way
to do that is to keep the same transfer function, except we are going to add a dead time to
it. So we'll have instead of one over tau(c) s plus one we'll have e to the negative theta
s divided by tau(c) s plus one. So what we'll now do is reevaluate the controller transfer
function using this new transfer function. So therefore the first term will still be
the same, we'll have one over k e to the negative theta s divided by tau s plus one. But our
second term will be a little different because we'll now have these e to the negative theta
s terms. So what you'll notice here is the fact that the e to the negative theta s that
was going to be so problematic before now cancels out. So what we now have is, we'll
invert the first term here, so we'll have tau s plus one over k, this is what our controller
transfer function is. We'll leave the one over tau(c) s plus one in the denominator,
and what we'll then do is find a least common denominator for the second term. And now we'll
have another cancellation with tau(c) s plus one. We have a slight problem right now in
the fact that we have this e to the negative theta s still in our controller transfer function,
and PID controllers do not have tuning parameters with exponentials. So therefore what this means
is the fact that we're going to have to take advantage of some sort of approximation in
order to do this. The podia approximation can be available, but here what we'll do is we'll take
advantage of an approximation which utilizes the Taylor series, and we can say that e to
the negative theta s is approximately one minus theta s. So when we substitute this
in we'll end up with that gc equals tau s plus one over k multiplied by one over tau(c)
s plus one, minus one minus theta s. So when we look at this we'll see that the ones cancel,
and that our controller transfer function is tau s plus one over k, multiplied by one
over tau(c) s plus theta s. Or we can state this as tau(c) plus theta multiplied by s.
So what we now need to do is get this into PID form, and the best way to do that is to
separate the terms. So here the s' will cancel out, and if we recall the form for a PID family
controller it will be kc times one plus one over tau(i) times s plus tau(d) times s.
As we look at this we have a term which is just a constant, that will be the p part of
the control, and then we have one with a one over s, and that will represent the i control.
So this will be a PI controller, so therefore to get the constant to be one we have to factor
everything else out. So this will be tau over k times tau(c) plus theta, and that will be
one plus, in order to factor appropriately the term with the s in it this will one over
tau. So therefore the taus would cancel and I would multiply out because we have the same
denominator of k times tau(c) plus theta. So this is our controller. So we have a PI controller
with kc equaling tau over k times tau(c) plus theta, and tau(i) just equal to the time constant
of the FOPTD. Tau(c) can be determined by a number of ways, there are a number of correlations
that have been developed, most of which are a function of the g transfer function, so
in this the first order plus dead time transfer function, and the important take home point
of this is the fact if you have a physically unrealizable controller using controller synthesis
the way to usually fix it is you have to alter your response, most likely to make it something
that is more reasonable and consistent with the transfer function which describes the
process, the sensor/transmitter, and the valve. And that is what we did in this screencast
was to take a look at how controller synthesis can be changed just by altering the process
transfer function to handle some problematic situations.