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‫Welcome back.

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‫So now that we have our equations of motion in the inertia frame, it's time to start looking at the

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‫feedback lionization controller.

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‫And first of all, what is a feedback controller anyway?

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‫Let's say that you have a random plant here and you also have a random state vector and the random reference

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‫vector right here.

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‫And therefore a random error vector would be our one minus X one, then R two, minus X two and our

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‫three minus X three.

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‫It makes sense, right?

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‫In other words, you can write it down like this.

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‫You have error one, error two and error three.

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‫So that's how it would look like.

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‫And so the states come out from here.

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‫Here you have the reference vector and then you feed your states here where you do the deduction reference

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‫minus the states, then you get your error vector and then in the feedback controller you have a row

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‫vector that is called K that looks like this K one, key two and key three like this.

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‫And so this K row vector, it's going to be right here.

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‫And then you have your control input.

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‫You hear mathematically you will write this, you vector like this you equals then this roll vector

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‫that contains these three key elements and then you multiply that by the error vector, which is a column

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‫vector like this.

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‫So you multiply your Rocchi vector by your column error vector.

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‫And if you do that then you will get K one times E one plus K two times E two plus K three times E three.

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‫Note that this is a scalar quantity here.

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‫It makes sense right here you have one by three, which is the dimension of this row vector.

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‫Here the dimension is three by one and so the inner dimensions cancel out and you're left with a value

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‫which is one by one and the one by one vector is a scalar.

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‫So it's similar to a proportional controller.

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‫Only now you take all your relevant states, you compare them with the reference state values and then

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‫you have multiple errors and multiple K values.

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‫And in the end you have a control input you, which is a scalar value, and then you find the key values

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‫that drive the error to zero.

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‫So your objective is to get error to zero.

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‫If this is your error dimension and this is your time dimension here, then you want your errors to

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‫go to zero like here as time goes to infinity.

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‫So mathematically you can write it like this limit.

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‫As time goes to infinity, your error should approach zero.

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‫That's your goal.

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‫That's the entire purpose of control engineering.

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‫So that's the general structure of the feedback control.

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‫And now let's see how it all fits in in our case.

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‫Thank you very much.