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‫Welcome back.

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‫And so this is your time sample line, and at zero, you have your present state a zero.

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‫Like I said, you don't predict that, you already know it.

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‫And then using this formula, you can predict your future state.

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‫For example, if you predict X one, then you will predict what the state will be in zero point one

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‫second.

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‫And then using the same formula, you can also predict your state at to your state at three and your

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‫state at four.

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‫And you don't need to do more because your horizon period is four.

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‫So your last predicted state would be X of four, so this one would be zero point two seconds into the

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‫future.

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‫This one zero point three seconds into the future and this one zero point four seconds into the future.

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‫So let's say that if you want to compute X for this predicted state zero point four seconds into the

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‫future, then the formula for that would look like this.

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‫You OK in this formula would be K equals four.

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‫So X are for equals then A to the power of four.

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‫So it's like multiplying for a matrices like this then that you multiply by your present state.

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‫And then in this vector you have first of all, this term A to the power of K minus one.

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‫So that would be a to the power of four minus one.

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‫That would be three then times the B matrix.

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‫And here you would have A to the power of two because K minus two, four, minus two, that would give

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‫you two times B, then here you would have A, B, so K minus three, four, minus three equals one.

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‫So you would have like one year and then finally you would have B here.

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‫Well actually you have A here but then you would have zero here because K minus four would be four,

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‫minus four which is zero and then eight to the power of zero would be one.

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‫So you can just say that it's one and then here you would have your control input at zero.

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‫This one here, then this one here will go here, then use up two would be this one and finally use

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‫up three would be this one here.

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‫And that was the way to directly compute except for without computing except one except two and except

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‫three first.

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‫And so if you write it out this entire formula, then it will look like this.

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‫And so the state vectors here are the following.

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‫This one is like this.

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‫So five zero five zero three zero zero zero and besides zero, that's how it looks like.

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‫And then this state vector looks like this.

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‫That's how it looks like.

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‫And of course, both of them, they are transposed because they are column vectors.

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‫And so that is what you predict them to be in zero point four seconds.

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‫Now, remember, this is a prediction.

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‫Your model does not take everything in the world into account.

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‫You USV will have true states at that time and those true states will be slightly different than these

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‫predicted states.

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‫And then your control input vectors will be the following.

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‫So you really know that this is your use of zero.

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‫It looks like this you one looks like this.

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‫You looks like this and then you three looks like this, and again, they're all com vectors.

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‫And again, to reiterate the A and B matrix, here they are, the discrete ones, OK?

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‫They're the ace D and B, the matrices.

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‫However, now we would like to compute the predicted X one, x two, x three and X four.

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‫We would like to compute these predicted states in one go and for that we created a more compact way

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‫to do it like this.

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‫This is a screenshot from the autonomous vehicle course.

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‫And you can see that here when you put your equations in this form, then you can compute all your predicted

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‫states in our case from one to four in one go for as long as you have your control inputs from zero

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‫to will.

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‫In our case, it was from zero to three and four.

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‫As long as you have your present state, then you can compute your predicted states in one go.

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‫And so we can do the same thing with the oveI, if that's the equation in order to predict your X1 x2.

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‫X three and finally X four, and we don't need more because our Horizon period was for.

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‫Then you can rewrite these equations in this form here.

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‫So if you write out these equations, then you will get the white ones here.

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‫It's exactly the same thing, only in a different format.

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‫And remember these A's and B's here they are sub matrices.

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‫So this is one huge matrix.

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‫And inside this huge matrix, you have sub matrices.

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‫So this is a sub matrix is a six by six sub matrix, and this B is a six by three sub matrix.

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‫That means that this entire thing here, it would be six times four.

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‫So that would be twenty four by six.

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‫And this huge matrix here, it would be twenty four by twelve, so twenty four rows and twelve columns

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‫because once sub matrix has six rows and three columns and here you have six rows and six columns and

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‫the number of columns does not expand.

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‫However the number of rows does expand.

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‫So twenty four rows by six columns.

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‫And one final thing.

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‫In our case you cannot have a predicted state that is larger than except for you cannot have X of five

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‫because remember then after X of four, you will already have a new outer loop.

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‫So the feedback lionization controller will give you a new target to chase.