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Let's listen.

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We have discussed how to design a robust controller as you, so it was pretty effective and gave us

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what we wanted, namely robustness against disturbance and parameter uncertainties.

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But the serious issue was the chattering problem due to the UM deteriorating controller algorithm in

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discrete domain or executing the control algorithm in discrete domain.

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We have said that chartering caused problems like stress on the structure of the robot due to high frequencies

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on model dynamics of the system can be triggered and also noise can be created and so on.

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So we wanted to avoid that.

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And in this lesson, we will see how we will avoid chattering or minimize it.

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So a how to our chattering problem?

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We will discuss two methods, namely boundary layers, which simple and dicey or discrete integral control

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method, which is more advanced and effective.

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Let's start with boundary layers method.

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What is the logic behind this method?

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We have discussed in last lesson that chattering was due to them switching like action control.

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So if we make this continuous control continues, then we can eliminate chattering.

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So how we will do that?

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The steps to apply boundary layers methods are simple.

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First, we have to choose boundary across sliding surface.

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We are doing that in order to categorize input control in two categories.

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Namely, we will apply discontinuous control.

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We have seen before outside the boundary region and apply continuous control proportional to the volley

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of sliding surface inside boundary region.

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In Equation C or IPSE is positive no, which designates width of the chosen boundary layer.

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Let's see with pictures this method, so it will become clearer.

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As we have said before, we will choose boundary layers across a sliding surface.

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The width depends on us and we can configure it.

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And here I write again the continuous control formulation or proportional control formulation.

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This is the input control graph with respect to the sliding surface.

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As you can see, green sections are discontinuous control because states are outside the boundary.

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Layers, which are designated by yellow dashed lines and red sections indicate when states are inside

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bounds rules.

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As you can see, red control part continuous road red control part is continuous control action.

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So by applying such control action, we can get this result.

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And as you can see, if we can reduce and almost near the chattering perfect, let's jump into the second

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method name this integral control method.

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This, called integral control, is more advanced and better than boundary less control because in boundary

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layers control method.

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While you can reduce chattering, your output variable doesn't completely reach the desired point,

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so you have steady state error.

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We can achieve both almost zero steady state error and avoiding chattering by using the IEC method.

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We will use the in this method.

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The condition that we have seen before but didn't utilize this condition is condition B.

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Namely, the rate of disturbance has to be bonded.

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We will see what it is.

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Let's assume that in our example, dynamic system, in our example dynamic system, both conditions,

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namely A and B, are met for the disturbance.

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Then we will develop the algorithm by just adding two terms to the basic VSC control.

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I mean, Barabbas structure control equation.

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We have seen one of these terms is proportional to the sliding surface, and this is like proportional

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controller, which we had also in bond related control.

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In both related control input was proportional to the sliding surface also.

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Second term is integral term, which is the solution for the steady state error we have seen in the

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bonds release method.

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This is the oral Formula C till the term is called integral control term because its derivative defined

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like that.

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From here, if you try to find C tilde, then we will get this integral to know in order this algorithm

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to work.

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There are some conditions in terms of coefficients of the control equation.

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Firstly, K has to be greater than zero by the year.

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I want to note wanting, as you know, before we have to choose this again m this k matrix this game

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or on key metrics.

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Excuse me this gain.

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It's not Matrix, okay?

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This game bigger than the magnitude of the disturbance and this caused increase in chattering disturbance

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was high.

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However, in this case, we don't require to choose as high as in previous case because see yield,

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the term is like disturbance estimate there and it will help to eliminate disturbances.

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So Kate can be chosen low and this is all an improvement in terms of reducing chattering, then h has

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to be greater than the rate of the disturbance.

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As C tilde is like estimates of the disturbance, it has to be sure the faster than the disturbance

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and the less condition.

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I don't know why it's like that.

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Just know that it should be like that or.

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And you may also read resources.

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Um, which I didn't hear.

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I want to note another thing.

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Also, this condition is valid for continuous disturbances, not discrete ones.

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Because when disturbance signal is discontinuous, it's derivative will be surely infinite, and age

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will always be less than infinity, always.

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So states will leave the sliding surface due to the discrete disturbance.

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They will see this exactly on simulation.

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However, if the disturbance is continuous, then we can choose age greater than the derivative of the

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disturbance, and the icy control method will successfully eliminate the disturbance and will be robust

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against the disturbances.

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Now let's go to the lab and examine the simulation results.

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OK, now let's check the discrete integral control in simulation, so I will do some changes in the

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lot.

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Indeed, some changes.

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Um, OK.

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Because it's it's almost the same code.

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There is nothing important changes, except some other animals.

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Okay.

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So um, as we have said, we will apply this grid integral control in order to reduce the effect of

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chattering.

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And let's see the, uh, what will be the result.

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As you can see, we will first, as we have said Vivo, we can take a little discouraged by using this

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individual control.

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We can take the key by both the game and I both am less so.

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As you can see before it was 150.

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Now we can take it as less as four.

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Yes, and we will choose the Lambda and H OK, and we will choose them as teenagers.

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300 OK, and let's do everything.

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I think other things are the same, so let's try to.

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Oh OK.

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And the most important thing, let's go our simulation and see what's um, what is the training as you

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as you know, we will change on the input control.

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OK?

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As you can see, we have added, uh, the proportional term integral term, OK, which we would need.

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OK.

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And um, um, and this is the standard control we have you before.

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OK, what we have, what we add here, in order to get P, we just multiplied, as you know, gain variable

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with this sliding surface.

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OK, we have calculated under four integral variable what we were doing.

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We were getting what we were doing.

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We were getting the sign of the sliding surface and we integrated, OK, as you can see, we have integrated

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it and multiplied y h.

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OK.

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And it was our integral term, OK, or C tilde term or, um, like, uh, disturbance observer.

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We will see what's this server?

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Scarlet's are on our program and see what's happening as you can see him.

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Um, anyway, you will not be able to see the chattering effect too much.

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I mean, it's a decrease or not.

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But the important thing is that, um, the program works as expected.

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And as you can see, we don't have steady state error at the end like in bounds release method, as

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we have said.

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As you can see, this is our sliding surface.

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As you can see, a sliding surface even changes in a much smoother way.

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OK, and not short.

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We light in bonder.

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Not like in when there is no chattering control method.

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OK, this is our input and it's interesting, as you can see in the first place.

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When we don't win, we are in the first place, it's something smooth, OK?

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Look to my, but it's smooth.

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If you make it in this way, you will see that it's almost smooth, OK?

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We have this discontinuity here, but up to here we have some smooth curve.

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And this is due to what this is due to.

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Until here we have one Typekit.

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I mean that it doesn't reach the sliding surface states doesn't reach the sliding surface.

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So we have combination of three signals, okay.

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And our input signal you either puts it is directed at a standard input standard input.

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You will it.

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This is totally you right here.

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But the standard input you, which which we have written as a sign of our Essex sliding servers, is

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in one direction.

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OK, so that's why we have only this one.

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So because this One Direction is combined with the integral term and Peter.

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However, when we reach sliding service, as you can see there in this case.

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OK, when the sliding surface, our standard control input arc gives us its switches.

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Um, OK.

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Between two values.

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But as you can see, it is not like before.

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OK?

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It's not something like 15 minus 50 200 minus 200.

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OK, it's not 200 minus 200 before it was 200 minus 200 OK, like it was switching from 200 to minus

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200 miles to 100 to 200, and it is a very high value.

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First of all, this is the problem.

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The second they eat is they are totally different values.

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So they were putting stress the with her.

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In this case, as you can see, it changes between almost 30 and 20.

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Yes, it changes between OK, served five, maybe or 31 may be OK, but we can say 30 and 20.

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As you can see, it doesn't change like 200 minus 200.

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It's very low volumes compared to before.

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OK.

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As you can see, and this is like, this is due to you.

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We have strip signals and these smoothness, as you can see, this curve is due to the um combined combined

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action of Integral Term and Peter, as you can see, and we have achieved what we want, we have reduced

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the effect of chattering very, very, very low degrees, OK, which we want.

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OK, let's see.

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No, let me first plug my computer into the charging before it turns off.

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So let's not add, uh, no disturbance, OK, some disturbance and see what's happening?

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OK.

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And I want you to compare the results in this one with the old also.

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OK, maybe you can rewatch the old video where we are running this, excuse me, a robust control or

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voice control.

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OK, remember, I will structure control without the RC and you can compare the results, but I think

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it's easy to see, uh, the difference between them.

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OK, now let's add some disturbance and see what's happening.

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Let's again make it more sporty.

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This sort of disturbance term.

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OK, and see, as you can see now, our disturbance term is what steps you on here steps.

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And as you know, steps signal is this continuous signal?

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OK, so what we have, this continuous signal will have derivative of almost infinity.

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OK.

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Because it changes instantly.

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OK.

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In a very, very um.

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Oh, can I say it?

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Change instantly?

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So that's why

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it's derivative.

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It'll be infinity.

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And also, as it's derivative, is infinity.

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Our h term will be always less than this for Delta One.

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OK.

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Don't forget condition B.

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So O states will leave sliding surface and DC control method will not be able to get rid of that and

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will not be able to return them back to the sliding surface.

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And let's see.

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OK.

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As you can see.

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Here, as you can see, our states come to the origin, but due to the.

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But due to the affected disturbance, they left the sliding surface OK.

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They left the sliding surface.

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They reached the sliding surface, OK, but due to the they reach the sliding service in less than one

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second.

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But we applied the disturbance in one second.

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OK, so that's why they left the sliding surface.

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OK had left the origin and came back again.

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OK.

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So, um, the DRC control couldn't manage to keep them in the sliding surface.

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It's due to water.

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As we have said, due to age is less than delta warm.

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Always because of this continuous signal you can see in slide and surface, there is an oscillation

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like this, OK?

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We are in the DRC couldn't handle the problem, and it couldn't keep the states on the or this sliding

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surface.

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OK, so that's why they left this sliding surface, OK?

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As you can see, this is our input control, and as you can see, we have a change at the output also

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output, but y also the oscillation due to, um, due to the, uh, discontinuous disturbance signal.

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OK.

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And so let's see what will happen if we change to change this discontinuous signal to the subject,

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continues one.

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OK.

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If we change with the continuous one, you will see that we will get much more smooth signal.

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OK, now let's check weather what will happen with the continuous disturbance signal not discontinuous,

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but continuous, as we have said.

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And with this continuous signal line step signal, uh, before the sliding uh, the states will leave

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the sliding surface as the derivative of a disturbance will be infinite.

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So let's not create our um continuous signal.

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It will be sine signal with amplitude 25 and frequency of ten applied in one applied in second one.

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OK, so the simulation is two seconds and I mean two time units.

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I think it's seconds and it will be applied on the first second.

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OK, let's see what we will do that in order to create that signal that we will use first, we will

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define and that time with something called zero point zero zero zero one from zero to one.

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Then we will create four time series for the two time and range for the second signal.

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The first signal will be zeros and the second signal will be twenty five since ten times T two minus

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one.

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The I have done minus one because I want to start this sound signal from zero.

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OK.

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00:18:30,940 --> 00:18:39,550
So as the T2 is run, if I didn't put anything here, it will start from sine 10.

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00:18:39,640 --> 00:18:45,490
OK, but I want to start from sine zero because I want to start this sound signal from zero.

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00:18:46,000 --> 00:18:46,480
OK?

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00:18:46,750 --> 00:18:55,390
We will combine our signals and we will use time series and function in order to get our disturbance

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time series function because we will import this signal in seemingly using from work space and it needs

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00:19:03,640 --> 00:19:04,870
time series data.

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OK.

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00:19:06,440 --> 00:19:07,510
Um, OK.

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00:19:07,540 --> 00:19:08,310
Anything else?

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00:19:08,320 --> 00:19:11,920
I didn't change, and let's see what will happen with the continuous signal.

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00:19:12,610 --> 00:19:13,840
OK, let's check.

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00:19:16,870 --> 00:19:24,010
As you can see with continuous signal and it applied, but as you can see, the states didn't leave

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00:19:24,010 --> 00:19:27,610
the sliding surface, okay.

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00:19:29,140 --> 00:19:34,270
As you can see, there is nothing in it oscillation in sliding surface as before in this cold case.

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Opposite to this cold case, OK, in this continued signal in the states left the sliding service and

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this is over, as you can see.

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Input signal.

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00:19:44,060 --> 00:19:44,530
OK.

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00:19:47,460 --> 00:19:47,840
OK.

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00:19:48,340 --> 00:19:50,110
And I think that's all.

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00:19:50,110 --> 00:19:58,850
But before closing, I want to note one thing here in order to show you that we have calculated see

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this integral one, you know, to show you that the sea is nothing but a disturbance estimate that led

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us to our AM disturbance again and looks to our disturbance.

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Zero.

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00:20:13,780 --> 00:20:14,350
OK.

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00:20:14,920 --> 00:20:21,250
Let's do so in order to make it zero, let's just change it to zero again.

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00:20:21,250 --> 00:20:22,000
Like this one?

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00:20:22,060 --> 00:20:23,140
Let's make it in this one.

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00:20:23,140 --> 00:20:28,250
So all the signals will be zero or the signal will be zero.

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00:20:28,250 --> 00:20:32,410
OK, and we will not have disturbance and let's check what will happen.

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00:20:34,030 --> 00:20:36,970
And let's plot the integral to it's here.

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00:20:37,110 --> 00:20:38,910
OK, let's put this intermediate term.

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00:20:39,310 --> 00:20:43,090
Excuse me, and let's see what's happening with our integral term.

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00:20:44,380 --> 00:20:45,220
OK.

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00:20:46,120 --> 00:20:47,080
Nothing interesting.

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00:20:47,080 --> 00:20:48,150
Nothing interesting.

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00:20:48,260 --> 00:20:49,810
No, no, no, no.

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00:20:49,930 --> 00:20:50,230
Yeah.

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00:20:50,740 --> 00:20:52,600
This is our item.

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00:20:52,630 --> 00:20:58,530
And as you can see, it's approximately, uh, certi.

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00:20:58,780 --> 00:20:59,290
OK.

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00:21:00,040 --> 00:21:09,250
It's it reason why it's worth it, because we have seen this one because our desired point is to and

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00:21:09,250 --> 00:21:15,730
as there is not any disturbance, we have to just apply or, yeah, 30 or more than 30.

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00:21:15,730 --> 00:21:22,880
We have to apply at least 30 of input in order to get and to reach the desired point.

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00:21:23,200 --> 00:21:26,200
And that's why our disturbance is here sorted.

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00:21:26,410 --> 00:21:32,620
As you can see, in this case, integral term is like our is the observer, OK?

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00:21:32,620 --> 00:21:35,350
Disturbance Observer, OK?

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00:21:35,590 --> 00:21:45,730
Because of that, we can choose a little K, OK, because the disturbance is mainly are are avoided

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00:21:45,730 --> 00:21:49,120
by the integral term due to the integral term.

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00:21:49,450 --> 00:21:52,270
Let's apply some disturbance also.

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00:21:52,690 --> 00:21:56,790
Let's check what will happen if we will do in this way.

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00:21:56,800 --> 00:21:58,960
Let's do make ones.

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00:21:58,960 --> 00:22:04,210
This will at times times minus 40.

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00:22:04,360 --> 00:22:14,650
OK, this will make step signal like, OK, we will apply at one second minus 40 disturbance and let's

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00:22:14,650 --> 00:22:20,740
see what will happen in terms of this with our, uh, OK, as you can see due to this signal.

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00:22:21,040 --> 00:22:21,430
Mm hmm.

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00:22:21,460 --> 00:22:26,350
OK, if this continues to be the states lift the state space, excuse me.

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00:22:26,350 --> 00:22:27,430
Uh, sliding surface.

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00:22:27,430 --> 00:22:31,870
And it's not interesting for me, but the interesting thing is the item.

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00:22:32,110 --> 00:22:41,860
As you can see, item is reached to Valley of 70 y 17 because we have forty for the disturbance and

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00:22:42,070 --> 00:22:48,610
30 for this disturbance because we have to apply, certainly in order to get wider and we have to apply.

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00:22:48,670 --> 00:22:55,120
Additionally, 40 in order to remove the disturbance, as you can see integral term and the final volume

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00:22:55,120 --> 00:22:55,890
of 70.

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00:22:56,150 --> 00:22:56,740
It will.

305
00:22:56,980 --> 00:23:02,860
It did what it it observed the disturbance.

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00:23:02,870 --> 00:23:11,320
OK, so that's why C is nothing but still is nothing but a disturbance observer which help us to choose

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00:23:11,320 --> 00:23:13,140
a little cape, OK?

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00:23:13,390 --> 00:23:20,060
And also it removes their static arora's that I missed in the state.

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00:23:21,670 --> 00:23:22,100
OK.

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00:23:22,120 --> 00:23:34,700
I think that's all about chattering removal or reducing the effect of chattering with the start of this

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00:23:34,780 --> 00:23:35,890
great integral control.