In this screencast we will conduct a stability analysis for a process being controlled by
a PI controller. So the information is given on the screen, and we have a second order
process transfer function, a zeroth order valve transfer function, and a zeroth order
sensor/transmitter transfer function, and what we're trying to do here is determine
what range of K(c) and tau(i) will allow the controller to be stable. K(c) and tau(i) individually
each have effects on stability. So as K(c), the controller gain, goes up, stability issues
go up, so if increasing K(c) causes increase in stability issues why would you ever increase
K(c)? As K(c) goes up the speed of your response also goes up, so therefore there is a tradeoff
here between the performance from a speed standpoint of your response, and whether or
not you process will become unstable. Similarly for tau(i), there is the same effect, except
only that for tau(i) the issues occurs as tau(i) goes down. So as tau(i) goes down
we have the same issues where we have a better response in terms of the speed the response,
however, it is more susceptible to stability issue. So what it means is we have these two
things happening independently, but when we put the two together in a PI controller there
is some symbiosis between the two of them, which means that you can't develop tuning
parameters for a PI controller by doing the K(c) part by itself and the tau(i) by itself
because both of them are having similar or potentially opposing effects, so therefore
these need to be analyzed together. In order to conduct stability analysis we're interested
in the denominator of the transfer function, which can be represented by the characteristic
equation, which states that one plus the open loop gain is zero, so therefore that will
be one plus the transfer functions for the controller, the valve, the process, and the
transmitter. So we have a PI controller here, so therefore that will have the transfer function
K(c) times tau(i) s plus one over tau(i) s. The valve is given as a transfer function
of two, the process has a transfer function of one over s squared plus two s plus two,
and the transmitter has a transfer function of zero point five. In order to carry out
this analysis we have to find the least common denominator, so that will be tau(i) s multiplied
by s squared plus two s plus two. Our first term, the one, needs to multiplied by both of
those terms that were just mentioned in the denominator, and the second term already has both
of those, so all we have here is K(c) times two times a half times one, so that will be
just K(c) times tau(i) s plus one. We will get rid of the denominator and then we'll
factor out. So when we factor out we'll have tau(i) s cubed plus two tau(i) s squared plus
two tau(i) s, plus K(c) times tau(i) s, plus K(c) equals zero. If we group the s terms
together to clean this up just a little bit we'll have our characteristic equation. So
from here we'll do a Routh analysis of the characteristic equation, and recall that a
Routh analysis requires all coefficients to be positive, and all terms in the first column
of the array to be positive. So as we look at our system here we can see that from the
s cubed term and the s squared term we end up with the same result saying that tau(i)
must be greater than zero. Additionally if we look at the constant term we see that K(c)
must be greater than zero. The s term will not provide us much more information because
we already know that tau(i) must be greater than zero, which dictates the fact that the
two plus K(c) term must be also greater than zero, so if I look at the two plus K(c), that
would tell us the fact that K(c) would have to be greater than negative two, but that
range is superseded by the one for the constant term that K(c) is greater than zero. So we'll
construct our Routh array. For the terms in the characteristic equation we start the top
left with the highest power term and work ourselves in a somewhat zig zag fashion as
seen here. Because this is a third order transfer function the only other value we need for
our Routh analysis that will be of utility will be b(1). And b(1) is defined as a relationship
between a variety of constants that are in the original characteristic equation. So recall
that with a Routh array the first term that you start with, the one with the highest power
of s, gets the annotation a to the n. For this case here will be tau(i). So moving on
from that the a(n-1) term will be two tau(i), the a(n-2) term will be two plus K(c) times
tau(i), and the a(n-3) term will be K(c). There's one tau(i) in each of the terms in
this equation so we can cancel those out, and what this leaves us with is that b(1)
is going to equal two times two plus K(c) times tau(i), minus K(c), all over two. In
order for this process to be stable this value must be greater than zero, so what we now
want to do is evaluate this, and what we note here unsurprisingly based on our earlier discussion
is that the values of K(c) and tau(i) are coupled. We'll get rid of the two, cause if
we move it over to the right hand side it will go to zero, and we're left with two times
two plus K(c), times tau(i) is greater than K(c) if I move the K(c) over. If I divide
to isolate tau(i) I show that tau(i) must be greater than K(c) over, if I factor out,
four plus two K(c). This makes sense because tau(i) in order for it to be stable, has to
be larger than particular values, so therefore the fact that this is a greater than sign
makes sense. We bring back our original criteria from our analysis of the constants, and we
get our stability range here. So again the important point to note here is the fact that
the stability of this process is dependent on both K(c) and tau(i). So in this screencast
we showed how to assess the stability of PI controllers, and also to note and see, the interaction
effects that occur between the two parameters of a PI controller, K(c) and tau(i).