MARTHA GROVER: In this ScreenCast,
we'll discuss the concept of stability,
where by stability we will refer to
bounded output for bounded input.
And, we will discuss stability
in the context of a Closed Loop System.
In particular, the Closed Loop Blending Process.
So, as in the previous ScreenCast,
we will use flow rate w two as our manipulated variable,
and mass fraction of Species a as our control variable.
And, we will close the loop, such that we calculate
the value of our manipulated variable
based on the measured value of the control variable.
So, what we want to do is to figure out,
if we make a bounded change to our input,
which in this case is our set point for x,
do we have have a bounded response?
So, stability is important for a closed loop system.
It's not sufficient to show good set point tracking,
for example, but it is necessary.
We do not want a system that goes unbounded in x,
or a bounded change in input.
So, one particular question that we can ask and answer is:
for what values of Kp, our feedback gain,
is the system stable, is the closed loop system stable?
And, we can determine the stability of the system
based on the closed loop transfer function.
So, previously, we had calculated G closed loop.
So, ideally, we would want these to match.
We would x to match its set point value.
But, that's not going to be possible in practice.
In particular, to calculate the stability
of our closed loop system, we can write out
the characteristic equation, and in the characteristic
equation, we set the denominator
of our transfer function equal to zero.
It then turns out that the closed loop system here
will be unstable if the real part of s is greater than zero.
So, this is a general result, but we can calculate it
for our particular system by solving for s.
So, by observing this expression,
we can see that it could take a positive value,
or it could it could take a negative value.
And, if s equals zero, that would be kind of
the critical point when we go from being stable to unstable.
So, let's go ahead and solve for that cross over point.
So, this tells us that if Kp is less that 160,
then the closed loop system will be stable.
Whereas, Kp greater than 160
in the closed loop system is unstable.
So, let's go ahead and consider a value of Kp equals -1000.
We considered that in the previous ScreenCast.
Now, there's a reason that we consider
negative numbers here, we have a negative gain
on our process, and that is going to require then
that we have a negative gain on our controller.
THere's a physical reason for that.
Stream two coming into the tank does not have
any of Species a, according to our set point
and study state values that we calculated previously.
So, by increasing the flow rate of w two into the tank,
we are diluting x in the system.
So, if we are too low, in terms of our value of x
in the tank relative to the set point,
we actually need to turn down
the flow rate going into the tank.
So, that's what this negative gain
on the system is going to give us.
We can calculate what is the value of s
in our characteristic equation,
if we have a proportionality constant of -1000.
So, we can see here, that with a proportionality constant
in our controller of -1000, we have a value of s
in our characteristic equation that is negative.
So, the system should be stable.
One way that we can consider the effect of Kp
on our closed loop system is to make
what's called a root locus plot.
So, we just calculated the root
of the closed loop transfer function
by solving the characteristic equation.
So, the root locus plot allows us to visualize
how the roots of a system change as we change a gate.
The root locus plot, at the imaginary value of s,
versus the real value of s.
Now, in this particular case, we don't have
an imaginary component, s is always going to be real.
But, we can plot how the roots of the system
would change as we change Kp.
We can plot this cross over value here.
And, in fact, we can get any value, the real part of s,
for an appropriately chosen value of Kp.
We can set s to be anything and solve for Kp,
so, our root locus actually goes
from negative infinity to positive infinity here,
but for any value of Kp less than 160,
we will have stability in our closed loop system.
We will have bounded output in x,
for a bounded change in x's set point.