‫Welcome back.
‫Let's now create a matrix vector form for the predicted states, you know, that X1 equals.
‫And then this equation here, then you also know that X2 equals this equation here.
‫But now what I can do, I can replace this X1 with all this.
‫And because of that, I can rewrite this entire equation like this or like this.
‫Then X three equals this equation here.
‫But this X2 here, it can be replaced with this equation here like this.
‫And as a result, you can rewrite this entire thing like this.
‫Finally, X4 equals all this, but this x three here, it can be replaced with this equation.
‫And as a result, X4 equals all this here this entire equation.
‫And now I can take X1, X2, X3 and X4.
‫I can put them all in one global vector, even though they are vectors as well.
‫But here they are, like sub vectors inside one big global vector.
‫And then I can take this equation here.
‫Then this equation here.
‫Also this one here and also this one here, and I can rewrite all this in this form.
‫If I write all this out, I will get back the same equations that you see here.
‫For example, if I take the terms with zero, you have one here, one here, also one here and then
‫one here.
‫All these terms are represented here.
‫And then all the other terms, they are represented here.
‫So I have simply taken these equations and put them in this matrix vector form.
‫And so if you look at this form and you compare it with this simplified version, then you can see the
‫difference in the simplified version.
‫You don't update your matrices internally inside your PC box when you predict your future states there.
‫You're essentially assuming that a zero equals a one equals A2 equals a three.
‫So just to specify, as you go through the plant, every inner loop, every 0.1 seconds, you have a
‫new A0 matrix.
‫But when you predict your future state vectors X1, X2, X3 and X4 inside your LPV PC box internally,
‫then in the simplified version, you don't update your a matrix.
‫You simply say that your A0 equals a and then you use your A0 matrix everywhere.
‫You don't go through this tedious process where even in your prediction phase, you generate new a matrices
‫every inner loop you take your A0 matrix.
‫That depends on 5.0 v2.0 and omega zero global.
‫And then you stick with that and you use it everywhere.
‫Then in the next in a loop, you get your new A0 matrix and again, you use it everywhere.
‫And that's why you can use your powers here.
‫For example, this one here eight to the power of three.
‫So eight to the power of three.
‫That would be triple A..
‫That's because the matrices are the same.
‫They are your A0 matrices.
‫However, in the non simplified version, you take an extra step and you update your AA matrices internally
‫inside your LTV NPC box while you are in the process of predicting your future state vectors x one x
‫two x three and X4.
‫And that's why, in this non simplified version, you cannot use powers instead of eight to the power
‫of three.
‫You will have something like this a three times a two times a one.
‫You cannot use powers because now these matrices are different A380 to a one a zero.
‫They are not the same matrices.
‫The simplified version is easier to code, but the non simplified version is more complete when your
‫horizon period is very short and the variables inside the matrices change relatively slowly.
‫Then it's reasonable to use the simplified version, and in this course, we will do just that.
‫Our Horizon period equals four, which is a very short, and the drone is expected to fly close to hovering
‫position, meaning that it is not expected to do crazy acrobatics.
‫So we can assume that our A0 matrix is more or less like our A1 matrix, which is more or less like
‫a too matrix and which is more or less like a three matrix.
‫And so we will simply use A0 or simply a in our entire state vector predictions.
‫However, when the horizon period is relatively long, let's say it equals 10.
‫Then inside your LPV and PC box, when you make your internal state vector predictions there, you will
‫be predicting x one x two up until x 10.
‫So you will have 10 different a matrices A0 A1 up until a nine since the prediction period is considerably
‫longer.
‫It would not be correct to assume that, for example, a zero is more or less a nine over a longer time
‫period.
‫These matrices can become considerably more different.
‫That's when you should use this non simplified version.
‫In fact, in the course, apply control systems to autonomous cars.
‫360 tracking I use the non simplified version over their horizon.
‫Period equals 10, and it can be made even longer.
‫And that's why the non simplified LP, the NPC is used there.
‫And as you can see in that car course, even the B matrix is updated.
‫You see b0, B1, B2.
‫That's because in that car, of course, the B matrix also contains a variable that changes there.
‫And so that's when I have.
‫To say about this non simplified version of the LP PC controller, I simply wanted you to know about
‫this more advanced version in case you need to use it in one day.
‫Thank you very much.