Let's listen.
We have discussed how to design a robust controller as you, so it was pretty effective and gave us
what we wanted, namely robustness against disturbance and parameter uncertainties.
But the serious issue was the chattering problem due to the UM deteriorating controller algorithm in
discrete domain or executing the control algorithm in discrete domain.
We have said that chartering caused problems like stress on the structure of the robot due to high frequencies
on model dynamics of the system can be triggered and also noise can be created and so on.
So we wanted to avoid that.
And in this lesson, we will see how we will avoid chattering or minimize it.
So a how to our chattering problem?
We will discuss two methods, namely boundary layers, which simple and dicey or discrete integral control
method, which is more advanced and effective.
Let's start with boundary layers method.
What is the logic behind this method?
We have discussed in last lesson that chattering was due to them switching like action control.
So if we make this continuous control continues, then we can eliminate chattering.
So how we will do that?
The steps to apply boundary layers methods are simple.
First, we have to choose boundary across sliding surface.
We are doing that in order to categorize input control in two categories.
Namely, we will apply discontinuous control.
We have seen before outside the boundary region and apply continuous control proportional to the volley
of sliding surface inside boundary region.
In Equation C or IPSE is positive no, which designates width of the chosen boundary layer.
Let's see with pictures this method, so it will become clearer.
As we have said before, we will choose boundary layers across a sliding surface.
The width depends on us and we can configure it.
And here I write again the continuous control formulation or proportional control formulation.
This is the input control graph with respect to the sliding surface.
As you can see, green sections are discontinuous control because states are outside the boundary.
Layers, which are designated by yellow dashed lines and red sections indicate when states are inside
bounds rules.
As you can see, red control part continuous road red control part is continuous control action.
So by applying such control action, we can get this result.
And as you can see, if we can reduce and almost near the chattering perfect, let's jump into the second
method name this integral control method.
This, called integral control, is more advanced and better than boundary less control because in boundary
layers control method.
While you can reduce chattering, your output variable doesn't completely reach the desired point,
so you have steady state error.
We can achieve both almost zero steady state error and avoiding chattering by using the IEC method.
We will use the in this method.
The condition that we have seen before but didn't utilize this condition is condition B.
Namely, the rate of disturbance has to be bonded.
We will see what it is.
Let's assume that in our example, dynamic system, in our example dynamic system, both conditions,
namely A and B, are met for the disturbance.
Then we will develop the algorithm by just adding two terms to the basic VSC control.
I mean, Barabbas structure control equation.
We have seen one of these terms is proportional to the sliding surface, and this is like proportional
controller, which we had also in bond related control.
In both related control input was proportional to the sliding surface also.
Second term is integral term, which is the solution for the steady state error we have seen in the
bonds release method.
This is the oral Formula C till the term is called integral control term because its derivative defined
like that.
From here, if you try to find C tilde, then we will get this integral to know in order this algorithm
to work.
There are some conditions in terms of coefficients of the control equation.
Firstly, K has to be greater than zero by the year.
I want to note wanting, as you know, before we have to choose this again m this k matrix this game
or on key metrics.
Excuse me this gain.
It's not Matrix, okay?
This game bigger than the magnitude of the disturbance and this caused increase in chattering disturbance
was high.
However, in this case, we don't require to choose as high as in previous case because see yield,
the term is like disturbance estimate there and it will help to eliminate disturbances.
So Kate can be chosen low and this is all an improvement in terms of reducing chattering, then h has
to be greater than the rate of the disturbance.
As C tilde is like estimates of the disturbance, it has to be sure the faster than the disturbance
and the less condition.
I don't know why it's like that.
Just know that it should be like that or.
And you may also read resources.
Um, which I didn't hear.
I want to note another thing.
Also, this condition is valid for continuous disturbances, not discrete ones.
Because when disturbance signal is discontinuous, it's derivative will be surely infinite, and age
will always be less than infinity, always.
So states will leave the sliding surface due to the discrete disturbance.
They will see this exactly on simulation.
However, if the disturbance is continuous, then we can choose age greater than the derivative of the
disturbance, and the icy control method will successfully eliminate the disturbance and will be robust
against the disturbances.
Now let's go to the lab and examine the simulation results.
OK, now let's check the discrete integral control in simulation, so I will do some changes in the
lot.
Indeed, some changes.
Um, OK.
Because it's it's almost the same code.
There is nothing important changes, except some other animals.
Okay.
So um, as we have said, we will apply this grid integral control in order to reduce the effect of
chattering.
And let's see the, uh, what will be the result.
As you can see, we will first, as we have said Vivo, we can take a little discouraged by using this
individual control.
We can take the key by both the game and I both am less so.
As you can see before it was 150.
Now we can take it as less as four.
Yes, and we will choose the Lambda and H OK, and we will choose them as teenagers.
300 OK, and let's do everything.
I think other things are the same, so let's try to.
Oh OK.
And the most important thing, let's go our simulation and see what's um, what is the training as you
as you know, we will change on the input control.
OK?
As you can see, we have added, uh, the proportional term integral term, OK, which we would need.
OK.
And um, um, and this is the standard control we have you before.
OK, what we have, what we add here, in order to get P, we just multiplied, as you know, gain variable
with this sliding surface.
OK, we have calculated under four integral variable what we were doing.
We were getting what we were doing.
We were getting the sign of the sliding surface and we integrated, OK, as you can see, we have integrated
it and multiplied y h.
OK.
And it was our integral term, OK, or C tilde term or, um, like, uh, disturbance observer.
We will see what's this server?
Scarlet's are on our program and see what's happening as you can see him.
Um, anyway, you will not be able to see the chattering effect too much.
I mean, it's a decrease or not.
But the important thing is that, um, the program works as expected.
And as you can see, we don't have steady state error at the end like in bounds release method, as
we have said.
As you can see, this is our sliding surface.
As you can see, a sliding surface even changes in a much smoother way.
OK, and not short.
We light in bonder.
Not like in when there is no chattering control method.
OK, this is our input and it's interesting, as you can see in the first place.
When we don't win, we are in the first place, it's something smooth, OK?
Look to my, but it's smooth.
If you make it in this way, you will see that it's almost smooth, OK?
We have this discontinuity here, but up to here we have some smooth curve.
And this is due to what this is due to.
Until here we have one Typekit.
I mean that it doesn't reach the sliding surface states doesn't reach the sliding surface.
So we have combination of three signals, okay.
And our input signal you either puts it is directed at a standard input standard input.
You will it.
This is totally you right here.
But the standard input you, which which we have written as a sign of our Essex sliding servers, is
in one direction.
OK, so that's why we have only this one.
So because this One Direction is combined with the integral term and Peter.
However, when we reach sliding service, as you can see there in this case.
OK, when the sliding surface, our standard control input arc gives us its switches.
Um, OK.
Between two values.
But as you can see, it is not like before.
OK?
It's not something like 15 minus 50 200 minus 200.
OK, it's not 200 minus 200 before it was 200 minus 200 OK, like it was switching from 200 to minus
200 miles to 100 to 200, and it is a very high value.
First of all, this is the problem.
The second they eat is they are totally different values.
So they were putting stress the with her.
In this case, as you can see, it changes between almost 30 and 20.
Yes, it changes between OK, served five, maybe or 31 may be OK, but we can say 30 and 20.
As you can see, it doesn't change like 200 minus 200.
It's very low volumes compared to before.
OK.
As you can see, and this is like, this is due to you.
We have strip signals and these smoothness, as you can see, this curve is due to the um combined combined
action of Integral Term and Peter, as you can see, and we have achieved what we want, we have reduced
the effect of chattering very, very, very low degrees, OK, which we want.
OK, let's see.
No, let me first plug my computer into the charging before it turns off.
So let's not add, uh, no disturbance, OK, some disturbance and see what's happening?
OK.
And I want you to compare the results in this one with the old also.
OK, maybe you can rewatch the old video where we are running this, excuse me, a robust control or
voice control.
OK, remember, I will structure control without the RC and you can compare the results, but I think
it's easy to see, uh, the difference between them.
OK, now let's add some disturbance and see what's happening.
Let's again make it more sporty.
This sort of disturbance term.
OK, and see, as you can see now, our disturbance term is what steps you on here steps.
And as you know, steps signal is this continuous signal?
OK, so what we have, this continuous signal will have derivative of almost infinity.
OK.
Because it changes instantly.
OK.
In a very, very um.
Oh, can I say it?
Change instantly?
So that's why
it's derivative.
It'll be infinity.
And also, as it's derivative, is infinity.
Our h term will be always less than this for Delta One.
OK.
Don't forget condition B.
So O states will leave sliding surface and DC control method will not be able to get rid of that and
will not be able to return them back to the sliding surface.
And let's see.
OK.
As you can see.
Here, as you can see, our states come to the origin, but due to the.
But due to the affected disturbance, they left the sliding surface OK.
They left the sliding surface.
They reached the sliding surface, OK, but due to the they reach the sliding service in less than one
second.
But we applied the disturbance in one second.
OK, so that's why they left the sliding surface.
OK had left the origin and came back again.
OK.
So, um, the DRC control couldn't manage to keep them in the sliding surface.
It's due to water.
As we have said, due to age is less than delta warm.
Always because of this continuous signal you can see in slide and surface, there is an oscillation
like this, OK?
We are in the DRC couldn't handle the problem, and it couldn't keep the states on the or this sliding
surface.
OK, so that's why they left this sliding surface, OK?
As you can see, this is our input control, and as you can see, we have a change at the output also
output, but y also the oscillation due to, um, due to the, uh, discontinuous disturbance signal.
OK.
And so let's see what will happen if we change to change this discontinuous signal to the subject,
continues one.
OK.
If we change with the continuous one, you will see that we will get much more smooth signal.
OK, now let's check weather what will happen with the continuous disturbance signal not discontinuous,
but continuous, as we have said.
And with this continuous signal line step signal, uh, before the sliding uh, the states will leave
the sliding surface as the derivative of a disturbance will be infinite.
So let's not create our um continuous signal.
It will be sine signal with amplitude 25 and frequency of ten applied in one applied in second one.
OK, so the simulation is two seconds and I mean two time units.
I think it's seconds and it will be applied on the first second.
OK, let's see what we will do that in order to create that signal that we will use first, we will
define and that time with something called zero point zero zero zero one from zero to one.
Then we will create four time series for the two time and range for the second signal.
The first signal will be zeros and the second signal will be twenty five since ten times T two minus
one.
The I have done minus one because I want to start this sound signal from zero.
OK.
So as the T2 is run, if I didn't put anything here, it will start from sine 10.
OK, but I want to start from sine zero because I want to start this sound signal from zero.
OK?
We will combine our signals and we will use time series and function in order to get our disturbance
time series function because we will import this signal in seemingly using from work space and it needs
time series data.
OK.
Um, OK.
Anything else?
I didn't change, and let's see what will happen with the continuous signal.
OK, let's check.
As you can see with continuous signal and it applied, but as you can see, the states didn't leave
the sliding surface, okay.
As you can see, there is nothing in it oscillation in sliding surface as before in this cold case.
Opposite to this cold case, OK, in this continued signal in the states left the sliding service and
this is over, as you can see.
Input signal.
OK.
And I think that's all.
But before closing, I want to note one thing here in order to show you that we have calculated see
this integral one, you know, to show you that the sea is nothing but a disturbance estimate that led
us to our AM disturbance again and looks to our disturbance.
Zero.
OK.
Let's do so in order to make it zero, let's just change it to zero again.
Like this one?
Let's make it in this one.
So all the signals will be zero or the signal will be zero.
OK, and we will not have disturbance and let's check what will happen.
And let's plot the integral to it's here.
OK, let's put this intermediate term.
Excuse me, and let's see what's happening with our integral term.
OK.
Nothing interesting.
No, no, no, no.
Yeah.
This is our item.
And as you can see, it's approximately, uh, certi.
OK.
It's it reason why it's worth it, because we have seen this one because our desired point is to and
as there is not any disturbance, we have to just apply or, yeah, 30 or more than 30.
We have to apply at least 30 of input in order to get and to reach the desired point.
And that's why our disturbance is here sorted.
As you can see, in this case, integral term is like our is the observer, OK?
Disturbance Observer, OK?
Because of that, we can choose a little K, OK, because the disturbance is mainly are are avoided
by the integral term due to the integral term.
Let's apply some disturbance also.
Let's check what will happen if we will do in this way.
Let's do make ones.
This will at times times minus 40.
OK, this will make step signal like, OK, we will apply at one second minus 40 disturbance and let's
see what will happen in terms of this with our, uh, OK, as you can see due to this signal.
Mm hmm.
OK, if this continues to be the states lift the state space, excuse me.
Uh, sliding surface.
And it's not interesting for me, but the interesting thing is the item.
As you can see, item is reached to Valley of 70 y 17 because we have forty for the disturbance and
30 for this disturbance because we have to apply, certainly in order to get wider and we have to apply.
Additionally, 40 in order to remove the disturbance, as you can see integral term and the final volume
of 70.
It will.
It did what it it observed the disturbance.
OK, so that's why C is nothing but still is nothing but a disturbance observer which help us to choose
a little cape, OK?
And also it removes their static arora's that I missed in the state.
OK.
I think that's all about chattering removal or reducing the effect of chattering with the start of this
great integral control.