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‫Welcome back.

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‫Let's now create a matrix vector form for the predicted states, you know, that X1 equals.

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‫And then this equation here, then you also know that X2 equals this equation here.

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‫But now what I can do, I can replace this X1 with all this.

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‫And because of that, I can rewrite this entire equation like this or like this.

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‫Then X three equals this equation here.

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‫But this X2 here, it can be replaced with this equation here like this.

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‫And as a result, you can rewrite this entire thing like this.

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‫Finally, X4 equals all this, but this x three here, it can be replaced with this equation.

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‫And as a result, X4 equals all this here this entire equation.

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‫And now I can take X1, X2, X3 and X4.

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‫I can put them all in one global vector, even though they are vectors as well.

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‫But here they are, like sub vectors inside one big global vector.

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‫And then I can take this equation here.

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‫Then this equation here.

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‫Also this one here and also this one here, and I can rewrite all this in this form.

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‫If I write all this out, I will get back the same equations that you see here.

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‫For example, if I take the terms with zero, you have one here, one here, also one here and then

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‫one here.

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‫All these terms are represented here.

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‫And then all the other terms, they are represented here.

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‫So I have simply taken these equations and put them in this matrix vector form.

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‫And so if you look at this form and you compare it with this simplified version, then you can see the

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‫difference in the simplified version.

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‫You don't update your matrices internally inside your PC box when you predict your future states there.

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‫You're essentially assuming that a zero equals a one equals A2 equals a three.

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‫So just to specify, as you go through the plant, every inner loop, every 0.1 seconds, you have a

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‫new A0 matrix.

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‫But when you predict your future state vectors X1, X2, X3 and X4 inside your LPV PC box internally,

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‫then in the simplified version, you don't update your a matrix.

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‫You simply say that your A0 equals a and then you use your A0 matrix everywhere.

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‫You don't go through this tedious process where even in your prediction phase, you generate new a matrices

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‫every inner loop you take your A0 matrix.

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‫That depends on 5.0 v2.0 and omega zero global.

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‫And then you stick with that and you use it everywhere.

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‫Then in the next in a loop, you get your new A0 matrix and again, you use it everywhere.

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‫And that's why you can use your powers here.

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‫For example, this one here eight to the power of three.

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‫So eight to the power of three.

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‫That would be triple A..

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‫That's because the matrices are the same.

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‫They are your A0 matrices.

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‫However, in the non simplified version, you take an extra step and you update your AA matrices internally

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‫inside your LTV NPC box while you are in the process of predicting your future state vectors x one x

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‫two x three and X4.

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‫And that's why, in this non simplified version, you cannot use powers instead of eight to the power

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‫of three.

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‫You will have something like this a three times a two times a one.

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‫You cannot use powers because now these matrices are different A380 to a one a zero.

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‫They are not the same matrices.

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‫The simplified version is easier to code, but the non simplified version is more complete when your

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‫horizon period is very short and the variables inside the matrices change relatively slowly.

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‫Then it's reasonable to use the simplified version, and in this course, we will do just that.

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‫Our Horizon period equals four, which is a very short, and the drone is expected to fly close to hovering

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‫position, meaning that it is not expected to do crazy acrobatics.

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‫So we can assume that our A0 matrix is more or less like our A1 matrix, which is more or less like

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‫a too matrix and which is more or less like a three matrix.

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‫And so we will simply use A0 or simply a in our entire state vector predictions.

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‫However, when the horizon period is relatively long, let's say it equals 10.

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‫Then inside your LPV and PC box, when you make your internal state vector predictions there, you will

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‫be predicting x one x two up until x 10.

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‫So you will have 10 different a matrices A0 A1 up until a nine since the prediction period is considerably

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‫longer.

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‫It would not be correct to assume that, for example, a zero is more or less a nine over a longer time

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‫period.

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‫These matrices can become considerably more different.

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‫That's when you should use this non simplified version.

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‫In fact, in the course, apply control systems to autonomous cars.

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‫360 tracking I use the non simplified version over their horizon.

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‫Period equals 10, and it can be made even longer.

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‫And that's why the non simplified LP, the NPC is used there.

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‫And as you can see in that car course, even the B matrix is updated.

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‫You see b0, B1, B2.

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‫That's because in that car, of course, the B matrix also contains a variable that changes there.

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‫And so that's when I have.

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‫To say about this non simplified version of the LP PC controller, I simply wanted you to know about

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‫this more advanced version in case you need to use it in one day.

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‫Thank you very much.