OK.
They have made a general introduction to the robust control.
Let's listen now we will continue with one of the robust control methods, namely variable structure
control or a sliding mode control
sliding surface control, which is another name given you understand in this lesson why these names
are given to this method.
Indeed, there are several different robust control techniques, but we will see variable, um, variable
surface control, variable structure control, excuse me, or the access control method.
You can ask why this control method?
Because firstly, it's simple to implement.
So if you can get your goal with a simple method right to look for difficult ones, then it provides
effective robustness against uncertainties and disturbances, which is the reason we are looking for
a robust controller.
Finally, as it is effective and simple to implement, it can be applied in many practical cases.
The best way of understanding the access control method and how it works is by working or an example
by the control algorithms.
We try to control the dynamics of the system, namely obtain desired dynamics.
For example, until now, by developing control algorithms, what was our purpose to make trajectory
tracking aerodynamics?
Asymptotically stable, wasn't it?
So as we are working over system dynamics, let's take a dynamic system and work on it.
Here is a second order dynamic system with positive coefficient.
Here, I want to note that here X can be replaced with any other variable that we want to control its
dynamics.
For example, we can replace X with a E, namely tracking error.
So we will get tracking error dynamics and we will try to control it as we want.
In short, we will work on X, which is general and can be replaced with any other variable.
OK, before developing a controller for the system, we have to know stability properties of our system,
namely whether it is stable or unstable or asymptotically stable.
In order to do that, let's first write system in states based form so we can easily analyze its eigenvalues
eigenvectors and saw from these stability properties.
We will follow standard procedure for getting system into state based form.
So first, define X one and X two variables as this.
These two variables are the states of our system.
Then we can write x double dot in terms of X one and x two states.
Finally, we can write system in matrix form as this.
Here is the evolution equation of the states from states based formulation.
Export equals X plus b u.
It's very well known equation, and I think you have already know this from this equation, we can determine
a matrix and as we don't have input you v will take B matrix as zero from this perfect.
But we want to see the output.
But we want to see at the output.
But would we want to see?
Excuse me.
But what we want to see at the output of the system, which states do we want to analyze?
We want to analyze both X1 and X2 or we want to observe, not analyze, but we can say, observe OK
and put both X1 and X1, namely Mm X and X stood because we want to see evolution of the states so we
can write output equation in this way.
And as the states base out of the equation is like that, we can determine see matrix from here, which
is nothing but identity matrix.
No, we will draw vs plot of the system to analyze the evolution of the system.
States don't let faze blow terminology intimidate you.
It's very simple, but very, very, very useful tool to determine convergence and stability properties
of our system.
Let's drill and analyze phase plot of our system.
We will do it in MATLAB, which will help us to do calculations faster and getting more initiation by
plotting several things, as you will see in a minute.
OK, so let's start to analyze the face of his portrait of our system.
Um, let's do that first.
I want to explain to you what's our.
I want to explain.
Wants thing to you, and this this will be the variable and you will see what's variable eight.
And so that's why I will explain what's variable and let's see first before the anger and the phase
plot analysis of our system.
We know that we can know our system's stability characteristics based on its eigenvalues.
Yes, if eigenvalues have positive real parts, what will happen?
The system will surely diverge because our system will be explored and unstable.
If eigenvalues have negative real parts, then our system will converge.
It will be stable.
It will be very beautiful.
Everything to sing well anyway.
And if we have a complex eigenvalues with positive result that our system trajectories will oscillate
and explode if in opposite way, we have eigenvalues with negative at all parts but a complex them,
they will oscillate but converge to the origin.
OK, so let's see.
What's our eigenvalues?
What's the type of like in our eigenvalues?
OK?
What kind of trajectories our system can have?
We want to see this.
OK?
In order to do this, we will analyze the overlords of our system as you can be as you.
We have said this, our system, OK, and we want to see, but symbolically, our eigenvalues.
OK, so let's define, see and see as symbolic variables.
And this is our aim matrix.
We have seen this and let's calculate the a matrix, OK before before putting a value here.
Let's just see what's our eigenvalues so run section and let's see our The Matrix.
OK, that's our eigenvalues, OK?
As you can see, it is our argument is we have two eigenvalues values.
I have written them here clearly.
As you can see, if you see him, there is here over two, but it doesn't have here because what I have
done, I take it and put it in inside the square, so it becomes inside the square root.
OK, it becomes divided by four.
So um, OK, this very simple.
So this is our first eigenvalues.
This is our second diagonal.
Excuse me, let me put it here minus C and minus minus C here.
Okay, minus perfect.
And I have many things on my table, so that's why I read first.
I write very slowly.
Anyway, let's continue.
So now what we want to see now what we want to see and we will give first.
We will replace the C.
OK, it's two um values of that.
I will say, OK, we will just give first c that I built a value of eight.
Excuse me what I say variable a not variable value of a.
OK, you can do this saying if you can say variable because its value will change anyway, which is
positive.
Okay, so what will happen if let's start with months, it will be easier.
OK, let's see.
Is positive?
No, what we have is put it or put it with minus eight.
So this will happen if we put minus eight instead of C V will have plus a and plus eight.
Yes, and a positive number.
So what will happen here?
C squared over four plus a.
OK, this is positive number.
And if we calculate the square, it will be higher than C over two.
So what will happen?
This eigenvalue will be less than zero.
This is perfect because we will have stable h.
OK, a good value like agent value.
But what will happen in this case?
This will be higher than zero and their unstable and unstable yes, agan value.
Let's see whatever it is, will be higher than zero because XY squared over four plus eight will be
higher than surely with the square root will be higher than she'll see over two.
OK.
Um, see our tour.
And also we have addition here because this side is positive.
This side is positive.
So positive, positive will give us positive.
So we will have positive eigenvalue, which means that our system you are like a multi unstable and
we know that ROM on stable item value is enough to make our system unstable so our trajectories will
explode.
However, we know that we have one eigenvectors or one eigenvalue that's stable.
One eigenvalue that is unstable.
OK?
Vs what?
This negative, OK, when a is we when we have a minus a C is equals to minus eight.
OK, so maybe write it in this week, then c OK, excuse me.
C equals two equals two minus eight.
We will have one stable who was stable, not one unstable time by road.
I mean eigenvalues.
OK, so overall system is unstable system OK, because we have an unstable root.
OK, so one unstable argument.
So what's happening if we choose a ipse as a, then this will be both of these will be minus.
Here is the interesting thing.
Well, if our aim is less than C squared over four, then what will happen?
Then this square root itself will be positive and the less than what c over two.
So this eigenvalue will be positive.
OK, let's make it greater than zero and this will become unstable.
So in this case, also, this will be positive because our aim is less than C squared or for this will
be positive.
Square root on the square root will be a positive.
Positive plus positive will give us positive and you will have unstable eigenvalue.
OK, so our system will be unstable.
So when C equals to a, we will have the one variant, which is one unstable root, OK, and one unstable
root and overall system is unstable system.
Okay, that's perfect.
Not perfect.
I mean, it's a bad thing, but our system is unstable.
So what happens when a is and OK, let me write it.
A here is, we have said, is less than less than C a c times c OK.
We could start with C terms.
OK.
E360 OK, I sorta what's happening here?
Let me write it down here.
OK, let's write it.
You can understand A is less than C squared c c squared divided by four.
OK?
The bottom we have, we can have also c equals to.
Um, let's see.
Let's go to this one.
I'm to lazy.
C equals to a but a is less than if it is in the news.
C What's happening here is it is less than not less than by the greater the uh, greater than the C
squared over four.
What will happen?
It is greater than C squared or for them now this will be negative and the square root of negative number.
What will what will happen?
We will have complex number with positive real part.
OK, so our system will be, uh, greater than zero.
Real parts will be the eigenvalue will have real parts greater than zero, but it will be complex number.
OK.
And also, this road will be complex because of Y because this square is greater than axis control for
this will be negative and the square root of negative number will be complex.
So we will have unstable either lower in both case, but oscillator its trajectory.
OK, oscillate
three, possibly.
Three.
OK, in the same way we can write for this one, I can believe it.
Oscillate three.
I can rally.
OK, oscillator trajectory.
OK, OK.
One unstable root.
We have c this one key here.
Let me write it again was unstable.
Unstable root, unstable system.
Eh?
Here is oh, we write it, we write it perfect.
But I still say is great again.
C squared or for this will be oscillator three because of complex, um, eigenvalue OK.
Because of complex eigenvalue.
OK, we understood this.
We will have two modes depending on the value of C.
OK, we can have.
You either have unstable or we will be unstable.
This oscillatory behaviour or gate, depending on the value of a ibs-c.
By changing the value of C, it will be easier or will be months.
So let's see here what will happen in this case.
So let's draw now phase plots of our system after we have understood there was a like and values of
it.
OK, let's see.
We have.
OK, if you will define global eight, you will see that y we define it as global.
This is our C this.
Were a as you can safely choose as greater than C squared over four?
OK, we will choose it two Times Square, Dover four and IPSE as minus eight.
So we will also thrive is a but let's choose just minus eight.
What will happen with minus eight?
We will have one stable route.
We will have one unstable route.
Our our system will be unstable.
Overall, we will see this one.
This is our matrix.
OK, this is our dispense simulation time.
OK, let's and we here make a few good Android Holden because we will do multiple plots on a good because
we will start with different initial conditions.
As you can see the optimize the different initial condition process, OK by using four and we will define
our initial condition here.
This will be our X1.
This will be our X2 and then we will use in order to simulate our system.
So extort equals the apex system.
OK?
In order to simulate this, we use or the E-40, OK.
Here we will provide will provide function with the name of derivatives, which is defined here.
As you can see, it takes the state as input and time, and it will give output the derivative of our
states.
So it will take X and we will it will call it X and give export as an output, OK?
And now you can see why we define the as global because we will use it inside this function also.
So we got our export, OK?
So I mean, X1, uh, X1 and X1 and X2 saw X, X and X thought, OK, and we split them because first
column is X.
The second column is export and we plot our next plot.
As we have said, what was our latest payload?
Facebook was showing their evolving trajectory of our states.
OK, so our state's evolution trajectory?
OK, so that's why we plot our states with respect to each other.
So we respect x x with respect x one with respect to X two or X with respect to extort OK with the black
color.
And we also plotted the initial conditions in order to see from where we have started.
And here I am doing what it.
This is not too much necessary, but here I just draw my eigenvectors.
There is nothing complicated here, but you can analyze yourself also.
This is just I get the equation of a line for eigenvectors and I just throw them.
If you have any question about that, please let me know under comment, and I will surely answer to
you how I did this calculation.
But if I have a not good go to this because it will take otherwise a bit of longer time and the necessary
time.
OK.
Anyway, let's simulate our system and see a phase plot of our system.
Let's simulated.
And that's it.
This is our face plot of our system, and it can be seen a bit intimidating, but it is very simple.
So this is our two eigenvectors.
As you can see the red one and the blue one, the blue one correspond to the diverging or unstable eigenvalue.
I don't know if it's positive.
The Red Line belong to the stable eigenvalue, OK, namely the one which has negative real population.
So look at that.
Our trajectories, the trajectories, evolving trajectory.
It's very interesting, as you can see.
For example, let's take this one.
OK, let's take this one.
It will be much more easier for us.
As you can see both of eigenvectors effects to the evolution of the system trajectories.
So let's start from here.
As you can see, we started from here.
It's our initial condition.
As you can see, we're trying to converge to the center with this stable or this stable eigenvectors.
OK, I can make that correspond to this stable eigenvalue because as you know, if our system is stable,
we will converge to the center.
So if we are over there, so in the direction of red eigenvectors, we try to converge to the center.
However, with the direction of green, excuse me, blue eigenvectors, the diversion of the Infinity-V,
I mean, system trajectories.
OK, so as you can see, their motion mixed motion affects today they their effects are mixed and affect
the system trajectory.
So our system to really try to converge in the direction of red one by the blue one.
It has a higher magnitude, so it will make our trajectory to explode or we'll make our system unstable.
So our tragic, tragic as it tries to convert the center, but it explodes in the direction of blue
one.
You can see this in other items, other system trajectories.
Also, as you can see here, see our system tries to go in the direction of, uh, red eigenvectors
to the center because it want to converge here there.
Because the effect of the red eigenvectors, both blue eigenvectors will affect and make it to divert.
OK.
As you can see, the divergence is higher than the convergence.
Why?
Let's see this.
Why this is because of the values of lambda.
Let's see our land, the values we have.
I mean, eigenvalues.
As you can see, positive eigenvalue is higher than the negative I can love.
You saw unstable root has higher magnitude than the what stable root saw.
Our system will diverge in the higher rate than it converges to the center.
OK?
You can see that as you can see, the divergence rate is higher.
It's immediate.
They tried to diverge to the UM diverge in the direction of blue eigenvectors because the route with
positive real part is higher than the bruit with negative real path.
OK, so as you can see, overall, our system is diverging because as you can see, every trajectory
of diverging to the infinity if we increase the time and make it, let me see.
For example, two second, uh, let's see what will happen if not to second.
But let's see one second and you will see the trajectories will continue to explode.
Let's see, as you can see here they are explored.
They just got infinite.
They don't converge to there.
As you can see, they don't converge to the origin because our system is unstable.
Perfect.
Let's come back to the zero point five in order to have clean fuel under.
And OK, let me mention one thing.
Also, as you can see, we we stated that what will happen if we start from these initial conditions?
But I want to note three things also.
First of all, if we start from the origin, so our initial condition in zero zero, we will stay in
zero zero hour trajectories will not either convert, I mean, not convert.
I mean, they will not move.
They will not be an evolution.
Our system will stay.
It will stay there.
It will not go to anywhere else.
But what if our system is our system trajectories is on our initial condition is on blue eigenvectors.
What we have said in blue eigenvectors system trajectories will diverge.
So if we are on blue eigenvectors, OK, blue, I can make Terra if there is no disturbance or any estimation
in the variable or anything else in.
If the system is purely ideal, then our initial can.
If we start initial tension over this blue eigenvectors, our trajectories will diverge to the infinity
because this blue eigenvectors is unstable.
If we start from here, it will divert this side OK, minus infinity plus infinity.
OK, what will happen if we are over the red one?
If we are over the red eigenvectors, then our system and trajectory will converge to their origin.
OK, which we want because we want our system dynamics worlds to converge to the origin.
It will slide over these eigenvectors and they converge to the Orygen, OK?
And if we start from here, it will converge to the origin.
So diverge in the blue eigenvectors, converge in the red.
I can work there, OK?
So here is the important thing.
So we will choose as our sliding surface what's ours and what's sliding service?
Let me briefly say that our sliding surface is the dynamics that we want to impose our system.
The sliding surface is very important in VSS control because this is the dynamics we want to impose
to our system.
So we want to impose our system that try our trajectories, go over this sliding surface.
So what we want as we want asymptotically stability, we want to choose our eigenvectors as this eigenvectors,
a sliding surface.
So our trajectories will come to this and we will make our trajectories to come to this sliding surface
or this eigenvectors sliding service is not the same as eigenvectors, but in this case, we will choose
this eigenvectors as our sliding surface because when our trajectories will reach to this eigenvectors,
they will just slide over to the origin and they will converge, OK and converge.
OK.
You can even choose as a sliding surface these diverging items, but you don't want this as a sliding
surface because you don't want all your system to explore to diverge.
Because if your trajectories will reach this matter, they will diverge.
But we want a smooth convergence to the origin, so we will choose our this eigenvectors azaan lot as
a sliding surface.
So our trajectories, system trajectories will reach there and they will just slide smoothly to the
origin.
And another note here is that these eigenvectors are what intrinsic to the system.
OK, we didn't imposes them.
We just choose this eigenvectors as our sliding surface.
We will make our trajectories to come to this eigenvectors orbit because you can see the trajectories
we want to diverge, but we will make them to come to this eigenvectors.
So then there will not be any input, but they will converge themselves to the origin because the trajectories
when trajectories is over these red eigenvectors, they will just smoothly converge at the origin so
we will get asymptotic stability.
OK, so that's why choosing the sliding surface correctly is very important.
OK, now let's see one thing we have seen our characteristic, as you can see, if you say that is minus
if you will have one stable route and an unstable route and our system will be unstable and we have
seen that.
So let's make it our ace plus and see what's happening.
In this case.
We have to have an unstable oscillator, a system.
And as you can see, we have unstable oscillator, a system.
As you can see, our trajectories start from there and they just spin and diverge spin and thought,
as you can see, they are diverging.
Whole system is diverging.
We can make it more clear by doing increasing the time.
Let's do it two seconds and let's see what's happening with two seconds.
OK, as you can see, as you can see what's happening, our system now starts from here and early.
Like in the oscillator behaviour, they are diverging to the infinity.
They look, look, look, look, look and diverge.
OK, as you can see, they are trying to move and diverge.
If you increase time, let's do five seconds.
The graph will be more complex.
I mean, by complex, it will be, as you can see, see, let's see the you can see the spinning here,
OK, see our trajectories spin.
Its oscillator is spin and.
Diverged, see, this is the phase plot, you know, it is really a very important tool, as you can
see, by just looking this, we can see that how our systems, how our system, whether our system is
converging or diverging and while speaking, I tried to hypnotize you, as you can see by doing this.
Okay?
And um.
OK.
I think it's enough to understand what's happening here.
So let's continue.
So we have seen that the dynamic system we analyze has two modes depending on the value of seek or efficient.
Namely, if C equals to minus a, we will get this mode, which we will call Mod A, as we have seen
before.
In this mode, we have one stable and one unstable route.
But one unstable route is enough to make the whole dynamic system state's evolution trajectory to explode
or system to become unstable.
If it was C as air, then we will get this mode, which will which will be called mode B.
Here are two routes are not only stable, but also excuse me here two roads are not only unstable,
but also complex.
So we get oscillates from behavior.
As you can see, we can obtain two different modes by changing just one parameter, which is C.
We can change trajectory evolution at anytime we want.
By changing the value of C as we have done the huge power, we can use it to achieve what we want,
namely to make trajectories to go to the origin, so obtain asymptotically stable system from unstable
system.
So what steps we will take, we will take as control input as we can impose value on it.
Then we will switch C from E to minus E or in the opposite way based on some control low to get asymptotic
stability.
What is that appropriate?
Control low?
As we have said before, our sliding surface choice will be this eigenvectors of the dynamic system,
which was asymptotically stable.
OK.
If initial condition was, uh or on this eigenvectors, it would converge to the center, as we have
seen during MATLAB session.
We want to drive our trajectories or systems states to this surface, and when they will reach to this
surface, they will just slide smoothly over this surface to the orders without any additional control
input.
Because as we have said before, if the states are in one of the eigenvectors, they will continue to
evolve in this direction while either exploiting or converging depending on the eigenvectors sliding
service will be indicated by us, which is a function of states.
If we draw a sliding surface, the coordinate frame will be separated into positive and negative parts.
Meanwhile, in positive parts, the valley of SW will be greater than zero and in negative parts will
be less than zero.
You can check the sign of SW by just picking random acts and extort values from both sides of the graph
and plugging them into the equation of sliding surface and analyzing the sign of the output.
OK, then we can write such control input where C is minus a ren x times X is less than zero.
So as C is minus a m, the system dynamics will be in mode eight and if X s will be less than zero,
then C it will be chosen as a saw.
The system dynamics will be in mode B.
From this picture, you can see the regions where excess is positive and where excess is negative on
positive origins.
Dynamics will evolve based on mode A. and on negative reviews based on mode B.
Let's say an example.
Let's assume that we have started from that red dot initial Poland as we are in positive origin mode.
A meme excuse me, as we are in positive region mode e will be activated and trajectory will evolve
based on this method, which we have seen before.
OK.
Trajectory evolved and restarted their negative region.
What to do?
Surely they will switch to the mode B in mode B.
Trajectories will evolve in this way until they reach a sliding surface indicated by Dark Blue Line.
And as we reach sliding surface, which we have chosen as the stable eigenvectors trajectory will evolve
to the Orygen without any additional control input.
This is because we do not have any disturbance and eigenvalue is intrinsic to the system.
You can analyze this graph by yourself where we started from another region.
Perfect.
We get asymptotically stable system from unstable system with very simple control method.
And this control method is robust.
Why?
Because when there will be uncertainties or disturbances on the system, they will try to deviate trajectory
from sliding surface.
So Trajectory will enter another region and control input will change based on it and will try to push
the trajectory or system states to this sliding surface again.
This is why another name of this control lizard is sliding more control.
We will talk about this more detailed way in future lessons.
OK, now let's see how we will implement our variable structure.
Um, well, I have a structure controller um, in or slightly surface controller in MATLAB, OK, and
see how and see how we can simulate the system, as you can see in.
Um, while we are analysing this plot, we have used the Odyssey 45 in order to analyse our system,
but now we will not do that.
Instead, we will use seems similar enough in order to make our job easier.
OK.
So what we'll do, we will define initial condition as always.
You know this if you will define our C as you know this, we will define our A. And we will define our
C, OK?
And there you know what to see.
OK.
Co-auteur minus C squared over four plus a square root.
OK.
And this is the simulation time t equals the ten ten time units.
We will simulate our system.
OK, now we have every variable here, so we will simulate our system in T time units.
And this is our simulated file.
Let's see our simulation file.
It's very easy.
First, we have our state case.
State, state, space, we hear, as you can see, we have written our aim at three zero one zero,
see why we have done here, as you see, because as we have said, see is our input.
OK, now we have said that see our is our input.
So we will see, as you saw, we will take out sea valuable from a instead.
We will multiply it with X and we will make C X as input u.
So we will change our B instead, so we will give here one.
So when we Colquitt X plus b, you v will get the same thing as when C was there, OK?
If you analyze it, you will understand it's very easy.
Okay.
Uh, OK, because we cannot write C here, because if we let's think that we have done as before and
write C here, what will happen in this case, we we have to change C at every time iteration, but
in this case we will not be able to do the to do this.
So instead, we give ipse as input here and we see X, I mean as input here.
So we calculate X plus b u instead of just x OK by of see here anyway.
OK, I hope that you understand it.
So this is our system and as we have said at the output will be X and extort because we defined our
C as identity matrix.
OK, so we will see at the observer dotted both X and X thought here is X and years X stood with them,
looks the multiplex where we split it.
Our vector OK into two parts in order to have X here and X not here, OK?
And we combine these two together in order to give as an output to our workspace.
OK, then we will take the variables from here and analyze.
OK.
So as we have said, what we will calculate in our next hour, uh, how to say a sliding surface because
based on sliding surface, we will define our input parameter C.
OK, so let's see o.
As we have said, our sliding surface will need X and extort.
So we will.
Our what was our sliding surface is minus C x plus x thought OK.
As you can see, by doing this one, uh, we have calculated our
um, sliding surface of our sliding surface was minus the X Plus extorts and we have done it's minus
C x plus x thought view calculated our s and what was our input.
As you remember, we have said that our input C will be minus E if X is Sova gets here.
S OK, let me write it here.
This is s what we have.
It is our sliding surface.
OK.
This is our s.
We multiply this as with X OK and the Output S6, let me write it also here.
So s x OK or axis OK.
This axis, based on its sign here for here we get sign.
This will return minus one.
If it is negative, it will return, plus one if it is positive.
Zero otherwise, I mean, the event is zero.
OK.
Based on the sign of excess, we will define our h.
So if it is negative, you will have minus eight.
As we have said, C will be minus eight or if it is positive, ibs-c will be plus eight.
OK, perfect.
We get C here and there as an input.
We VE also write its value to the input, but we will add to the workspace, but we will not analyse
it because it's not needed.
Even we can delete this OK, because we will not need it.
OK.
Here is just our C valuable.
We multiply it with minus one because um yeah, we ve multiplied minus one because what was our state's
space equation?
It was in order to extort equals x plus b you.
But our what was our a matrix?
It was zero one minus c c OK.
We take out minus C from a matrix.
As we have said before, it was before their minds c because now it's input so of.
See here we multiplied minus once obviously will have minus C v, multiply it with X.
As you can see, we multiply it with minus X and we come we we m some it with a X, so we will have
the same effect as a equals to zero one minus six C.
OK, perfect.
So this is our state space with this our system.
OK.
And be it robust control, sliding surface control again.
And let's analyze the results.
What will happen.
So we will simulated and then we will get the outputs OK?
As you can see, we will get first time in order to plot.
We will get x um from the states.
As you can see, this was our state and we get, uh, data column vector first column vector, which
is first column, which is on the X and the second column, which is X thought, OK, then we will create
a future and evil plot X with respect to X in order to see what's happening with our, um, what's happening
with our stay trajectories.
So let's do that and it's very interesting for me.
The result?
Oh, as you can see before, what was our system before our system was diverging?
No, it's very interesting.
Look, what's happening.
This is our initial condition.
OK?
We are starting from minus four minus two.
As you can see.
First, our system is in in its approximately in region.
It is in the region.
I think it's in region four, OK, where the mode B is active, OK, and our system is what it is.
Um, it tries to embody what was that must be.
Both of our eigenvalues was negative real parts, but they were complex, so it will have oscillated
of behaviour also unstable.
So it will go like that our c, but they will reach to the our stable eigenvectors.
And as you see, as you can see it, how it comes.
I have said that it is when it reaches the island.
This is all like, OK, first, let me see this.
Our sliding surface, this stabilizing factor, as you can see, our trajectories reached two there
and they just slide to the Orygen C slide to the zero y zero, as you can see from an unstable system.
But we have to get asymptotically stable system.
Our trajectories come to this sliding surface and they just slide to the centre without any problems
smoothly.
OK, let's change our initial condition and let's see what we will get.
OK?
As you can see, we have no different characteristic.
OK?
Here is our initial condition.
It comes directly to the sliding surface and it just slide to the Odigem with the sliding surface over
the sliding surface.
It's very beautiful.
Yes.
So let's try to something else.
I don't know whether I will be able to do that.
Um, let's see what will happen if we choose four
to five.
OK, this is really interesting to see how it comes.
It came like that, OK?
Our system can load that and come to the eigenvectors or a sliding surface and directly slide it to
the surface.
OK.
So you can change the initial condition.
You can start from the different regions and you will see that at each and every time the system will
be asymptotically stable.
OK.
And here you can see the importance of sliding surfaces because if we didn't use sliding surface correctly,
we can by virtue OK, not converge, OK.
We have to choose our sliding surface carefully in order to just slide over these and come to their
origin.
OK.
And as you can see, this is a robust system it does not depend on from where you are starting your.
If you think this instead of here, we would have a road.
OK?
And then it does doesn't depend.
Where is our error or what's our initial error?
OK, it will come either the state trajectories will converge to this zero and way.
So that's why it is called robust controller and variable structure controller.
Because as you can see, we change the our input value.
OK, it's variable variable structure and we get different modes.
We get different trajectories, system trajectories and we reached two there.
And this is the origins of you get us into a stable system.
So that's why it's called a variable structure control.
And also, yeah, that's why it's called sliding surface control, because we are trying to make our
system trajectories to go to the sliding surface and with the sliding surface to the origin.
OK, if you have any problem with the code, please write me, ask me and ask me and I will try to answer
to you, OK?
Before concluding this list.
And let me mention some things to you.
Firstly, as we have said, sliding servers is nothing but the dynamics we want to impose to our system.
So steep trajectories will evolve based on these dynamics.
As you may notice, this makes us to choose the sliding surface very carefully to achieve our goal.
So now let's see briefly the important steps to design these as controller first and most importantly,
choosing appropriate sliding surface of ASX.
Then choose appropriate control to make state trajectories evolve or a sliding surface.
We will talk about this truly in the next few lessons here.
I want to note one important thing.
In previous example, eigenvectors was intrinsic for a dynamic system.
Namely, we didn't impose it to the system as a new dynamics.
We can understand that from this sliding surface is intrinsic to our system because in phase plots,
when we take our initial points on this IGen vector, they have solved the slide over it.
This is not correct when we start somewhere else or we impose some new dynamics to the system because
state trajectories will try to leave that sliding surface because it is not intrinsic to them and they
want to be free.
But you have to be a dictator and force them to go over the leaving surface you want.
Excuse me, or the sliding surface you want.
Let me know that the basics of robust control is developed by Soviet engineers, so you can understand
the logic behind this method.
Additionally, if we consider disturbances and uncertainties, these will also try to divide the system
trajectories from the sliding surface, as we have seen during introduction to the robust control.
So you have to apply additional input m to make system trajectory stay on sliding surface, even its
lower sliding surface.