‫Welcome back.
‫In the previous course, we also derived this formula here for being able to predict future states at
‫some kind of future time sample, we made these kind of substitutions where, for example, we wrote
‫X1 in terms of X zero and Delta zero.
‫Now, in the other case, we can do it exactly like we did it in the previous course by employing this
‫formula here.
‫The only difference now, of course, is that instead of your Delta being as your control inputs, you
‫have this vector as your control inputs.
‫So for example, you're you zero would be like this, you two zero three zero and EUFOR zero.
‫Now note that here the letter K has a different meaning.
‫Usually we say that if you have X, sub K, we say that it's a present state and X at K plus one.
‫It is a state one time sample in the future.
‫So zero point one seconds in the future in our case.
‫But in this formula, if you say that your K equals four, then let's say that you have X four instead
‫of X, OK, you have except for then here in this formula it means that this X four is the predicted
‫state vector four times samples in the future.
‫So this thing here, it would be now plus in our case, four times sample time intervals.
‫So that would be now plus zero point four seconds later.
‫So in the convention that you are used to, it would be X, sub K plus four.
‫Right.
‫But in this formula here, this key means how many times samples into the future you have predicted
‫your states.
‫You can also think of it like that.
‫So this is your state vector and that zero plus four or zero plus.
‫OK, so that's the convention with this specific formula.
‫So let's say that we want to predict the states four times samples in the future.
‫So this would be your now here.
‫This would be your X up zero.
‫So this is your present state except zero.
‫And that's why I said that you can think of it like this when you have this K, you can think of it
‫in this way, X up zero plus K because a zero means your present state in this formula.
‫And then the zero point one seconds later you will have your predicted state X one X up to accept three
‫and X for.
‫And also remember that your NPC horizon period is also four.
‫That means that the NPC finds control inputs, you zero, you one, you two and you three.
‫But the plant only receives this one, the first one and then it discards the rest.
‫However, if you want to predict your future states, right, because remember, these are the predicted
‫future states.
‫This is what you know, this is your present state and these are the predicted states and you're using
‫the mathematical model of your system in order to predict these future states into the future.
‫In order to do that, you do need to have these control inputs as well, because you have to plug them
‫in into this vector in order to be able to find your predicted future states like this one here, for
‫example, if you want to find except for then in this vector here, your control inputs must go from
‫zero up until three.