‫Welcome back.
‫So now that we have our equations of motion in the inertia frame, it's time to start looking at the
‫feedback lionization controller.
‫And first of all, what is a feedback controller anyway?
‫Let's say that you have a random plant here and you also have a random state vector and the random reference
‫vector right here.
‫And therefore a random error vector would be our one minus X one, then R two, minus X two and our
‫three minus X three.
‫It makes sense, right?
‫In other words, you can write it down like this.
‫You have error one, error two and error three.
‫So that's how it would look like.
‫And so the states come out from here.
‫Here you have the reference vector and then you feed your states here where you do the deduction reference
‫minus the states, then you get your error vector and then in the feedback controller you have a row
‫vector that is called K that looks like this K one, key two and key three like this.
‫And so this K row vector, it's going to be right here.
‫And then you have your control input.
‫You hear mathematically you will write this, you vector like this you equals then this roll vector
‫that contains these three key elements and then you multiply that by the error vector, which is a column
‫vector like this.
‫So you multiply your Rocchi vector by your column error vector.
‫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.
‫Note that this is a scalar quantity here.
‫It makes sense right here you have one by three, which is the dimension of this row vector.
‫Here the dimension is three by one and so the inner dimensions cancel out and you're left with a value
‫which is one by one and the one by one vector is a scalar.
‫So it's similar to a proportional controller.
‫Only now you take all your relevant states, you compare them with the reference state values and then
‫you have multiple errors and multiple K values.
‫And in the end you have a control input you, which is a scalar value, and then you find the key values
‫that drive the error to zero.
‫So your objective is to get error to zero.
‫If this is your error dimension and this is your time dimension here, then you want your errors to
‫go to zero like here as time goes to infinity.
‫So mathematically you can write it like this limit.
‫As time goes to infinity, your error should approach zero.
‫That's your goal.
‫That's the entire purpose of control engineering.
‫So that's the general structure of the feedback control.
‫And now let's see how it all fits in in our case.
‫Thank you very much.