So recall, last time, we saw that. We
designed a controller that was nice and
smooth. It didn't overreact to small
errors. made a system stable. Yet didn't
achieve tracking. And this was the
proportional regulator, or the p
regulator. and let's return to our
performance objectives a little bit. We've
talked about them briefly. But a
controller at the minimum should.
Stabilize the system. If it doesn't do
that, we know nothing and I've written
this rather awkward looking acronym here,
BIBO, which is something out of the Lord
of the Rings almost. What it stands for
is, bounded in, bounded out which means
that if the control signal is bounded, the
state of the system should also be
bounded. What this means is that, by
doing. Reasonable things the system
doesn't blow up. And our system doesn't do
that. Tracking means we should get to the
reference value we want. And robustness
means we shouldn't have to know too much
about parameters that we really have no
way of knowing. And preferably we should
be able to fight noise as. Well, so recall
at this was the model and when I
introduced this wind resistant term here,
we had a little bit of a problem.The
proportional regulator couldn't overcome
it and lets have another controller done
one that explicitly cancels out the effect
of the wind resistance. So here is my.
Attempt 3, I'm going to use this part,
which is the proportional part that we
already talked about, and then I'm going
to add this thing which is plus gamma
m/c*x.
Well why did I do this? Well, I did this
for the following reason that if you reach
steady state x is not equal to 0, then now
What you get is well this was the p part.
This is the controller, the p controller.
And then the effect of this thing well
you're going to multiply this by c/m. What
you're going get then is plus gamma x. And
then you have wind resistance which is
negative gamma x. So the gamma x, the bad
parts cancel out. And in fact all we're
left with then is that x. Has to be equal
to r. So, voila, we've sol ved the
problem.
We have perfect tracking. Or, have we?
dom, dom, dom. No, we have not. And, why
is this? Well, we have stability and we
have tracking, but we don't have
robustness. Here are three things that we
don't know. Gamma, m, and c. And our
controller depends explicitly on, On these
coefficients. So all of a sudden we have
to know all these physical parameters that
we don't know, so this is not a robust
control design. So Attempt 3 is a failure.
Okay, let's go back to the P-Regulator and
see what's going on there. What, what's
actually happening is that the
proportional error is doing a fine job
pushing the system up to close to where it
should be, but, then it kind of runs out
of steam, and it can't push hard enough to
overcome The effect of the wind
resistance. So the proportional thing
isn't hard enough, but take a look here.
This is the error, then the error starts
accumulating over time, so if we somehow,
if we're able to collect All of these
errors over time, even though they are
very small. Over time, that should be
enough, so that we can use this now
accumulated error to push all the way up.
So I wish there was some way of collecting
things over time in a plot like this. And,
of course, there. There is, this is
something called an integral. So, if we
take the integral over the error we're
collecting the error over time and over
time as this errors going to accumulate
it's going to give us enough pushing power
to actually overcome the wind resistance.
So attempt 4 is a pi. Regulator.
So what I have here is the error at time
t. This is my kp, which is my proportional
gain. So this is the p part that we
already saw. And now, I'm adding an
integral that is integrating up the error
from. The beginning to the current time.
And it's collecting this. And then we have
another term here, or another coefficient.
The ki, where I stands for the integral
part. So this a pi regulator. And it is
2/3 of. The most common regulator found
anywhere in the world, and in fact it's
almos t 2/3 of commercial grade cruise
controllers. So if I have a p and an i,
what could possibly be missing to get to
all of them? 3/3 instead of just 2/3.
Well, we take a derivative. Right, we have
proportion, we have integral, and we have
a derivative. So, why not produce what's
called a PID-Regulator? So now we have a
proportional term with a proportional
gain. We have an integral part with an
integral gain. And then we have a
derivative part with a derivative gain, so
this is. It's an extremely useful
controller that shows up a lot. And, in
fact, I'm going to hand, have to hand out
a big sweetheart to the PID regulator.
Because it's such an important type of
control structure that shows up all the
time. And in fact we're going to get quite
good at designing the PID regulators. Now
having said that, I can draw hearts all I
want, let's see it in action and see what
it actually does. And if I use just the PI
regulator, not even a D component to the
cruise controller, then all of a sudden I
get something that's getting up quickly,
nice and slowly, I mean smoothly, to 70.
Miles per hour, which is my reference. So
this solves the problem. I don't know
parameters, so it's robust. I'm achieving
tracking, because I'm getting to 30 miles
per hour. And, I'm stable in the sense
that I didn't crash. So, this seems like a
very useful design.
EmoticonEmoticon