‫Welcome back.
‫I now want to caution you about something, suppose that you have a differential equation like this.
‫This one here, and let's put it in the state space form.
‫So your X dot equals X dot and then your X double dot equals minus two times X minus three times three
‫times X dot and then plus sine T like this.
‫And so we will have our X don't here X double dot here you will have your A matrix here and then you
‫will have your state vector X and then X dot and then plus you will have your B matrix here times sine
‫T..
‫So here you will have zero and one because X equals X dot and then here you will have minus two and
‫then here you will have minus three times T and then in the B matrix you will have zero here and one
‫here.
‫So you can see that this differential equation here, it's linear, but it's linear time variant, so
‫it's LTV.
‫That's because you're a matrix.
‫Here is a function of time.
‫It's an LTV system.
‫Let's now find the systems, poles or eigenvalues.
‫And remember, the eigenvalues are found like this.
‫You take this a matrix, and then from that you will subtract a matrix where you have these eigenvalues
‫like this, and then you take the determinant of it and then you equate it to zero.
‫So the difference of these matrices will be this one.
‫And then you take the determinant of it.
‫And again, you equate it to zero.
‫And so now you will have minus lambda times, minus three times T minus lambda and then minus one times
‫minus two.
‫So that's how you compute the determinant.
‫This element times, this element minus this element times this element and that equals to zero.
‫So if you write it out then you will have three times lambda times T plus lambda squared and plus two
‫equals zero or in other way you can write it like this lambda squared plus three times T and then that
‫is multiplied by lambda plus two equals zero.
‫And so your lamda and you will have two of them.
‫It will be minus three times D, which is this one here, divided by two plus minus, and then you will
‫have square root where you will have three times tea over to you, square it and then you subtract two
‫from it.
‫So that will be your Lunda one.
‫And that will be your lamda to hear.
‫As you can see, your polls now are not close to numbers.
‫They are a function of time.
‫You have the time variable here and here.
‫So your loved one is a function of time and then your number two is a function of time as well.
‫So what does it mean?
‫Well, it means that the polls are not fixed in the Laplanche domain.
‫These polls, they move with time, they are not in the same location all the time.
‫So with time, the polls can become complex.
‫Or one of them can become positive or both of them can become positive.
‫And that's not good.
‫And things get even worse when you have non-linear differential equations.
‫Suppose that you have this differential equation here and you see now here, you will have X squared.
‫Let's put it in the state space for, again, your X dot equals X dot and then your X double dot equals
‫minus two times X squared, minus three times T times X dot and then plus D squared.
‫So here you will have X dot and then X double dot.
‫Your state vector is X and X dot.
‫Your input function is T squared, and so here you will have zero and one, and now here, since you
‫have this X squared here, then you will have minus two times X here and then minus three times T here.
‫So if you write it out, then you will see that minus two times X, you would multiply it by X and then
‫you will have X squared.
‫And then here you will have zero and one.
‫So you can see that this is a nonlinear system and let's find the Poles now.
‫So this difference here, this matrix minus this lamda matrix, and then you take the determinant of
‫it and then you equate it to zero, so you will have.
‫This matrix, and then you take the determinate of it and you equate it to zero.
‫Taking the out of it will give you this expression.
‫After some mathematical manipulation, you will have this expression here.
‫And then your polls are in this form, one with a plus sign and one with a minus sign, so you can see
‫that your polls now depend on time and also on the state itself.
‫Which in turn depends on time as well.
‫So I can write it down like this, No one is a function of time and also a function of the state, which
‫itself is a function of time.
‫And the same thing for London, too, so again, it's not fixed.
‫Again, the polls can become complex or positive or both with time like these polls here, they can
‫move around, they can go like this closer to each other, and then they can become complex like this.
‫And then both of them can even move here and then they can be complex and unstable.
‫So your system might start behaving very unpredictably, so it might start oscillating, become unstable,
‫and a lot of trouble can come from that.
‫So my point is that this Paul Placemen method should only be used when you have an LTI differential
‫equation.
‫In other words, if you put it in the state's space form, X dot vector equals eight times X vector
‫plus B times and then the input vector.
‫Then you're A and B matrixes, they should be constant, they shouldn't vary with time, and they shouldn't
‫depend on other states either.
‫They should be constant matrixes.
‫Then your polls will be fixed in place and you can rely on this method.
‫And now, if you remember then in the previous section, when we dealt with more predictive control,
‫then we in fact had this situation, our matrix was not constant.
‫In fact, it had states in it.
‫And we called that method linear parameter, varying method or LTV.
‫And so what I want to tell you now is that LTV is a robust control method.
‫There are books written about it and essentially what l.P is doing.
‫In the A and B matrix, it is treating those variables that are not fixed.
‫It is treating them as parameters.
‫Let's say parameter one or parameter two, and then also here, parameter one and parameter two, and
‫those parameters, they consist of these variables that are not fixed in the A and B matrices.
‫In this case, only a matrix.
‫And essentially, the linear parameter varying theory as part of robust control is trying to determine.
‫How much these parameters can vary in order to have acceptable results?
‫So just as a quick example, let's say that you have two axes here and then here you will have X and
‫then here you will have T, the two variables that vary in this matrix.
‫And then this linear parameter varying theory tries to determine the regions.
‫For these varying parameters.
‫So, for example, if you're somewhere here, so your ex is here and then your time is here and so you
‫are here, then that's OK.
‫But then if you're somewhere here where your ex is kind of OK, but then you are out of this box because
‫your time has varied too much.
‫So in the end, you are in this region here, then that's not OK because you are out of this region.
‫So there is an entire theory about it, but this is a very initial superficial introduction to it.
‫In our case, in our course, for the trajectories that we have, this LPT method works very well.
‫And the variables that are inside our NPC, a matrix.
‫That, if you remember, was a six by six matrix.
‫It contained five dot and theater dot and then also Omega.
‫In our case, these varying variables, they didn't vary enough.
‫To ruin the performance of our drone, but this OPV method, this is definitely one of the methods that
‫you should study if you want to become more advanced in control and perhaps focus on robust control.
‫And so in the next video, we're going to focus more on our specific feedback lionization controller.
‫We're going to start wrapping things up.
‫Thank you very much.