‫Welcome back.
‫Now, if you remember from the previous course, then we really wanted to keep the changes of control
‫inputs as small as possible.
‫And I'm talking about the changes of control inputs, the deltas, the increments.
‫So if you remember, we modified our cost function in such a way that in the vehicle case, we started
‫treating the increment of the steering wheel angle, Delta Delta at K plus I as our control input,
‫but we had to keep track of the real steering wheel angle as well.
‫For that we define our Delta Delta at Key like this, Delta at K, minus Delta at K, minus one.
‫And so we decided that we would keep Delta at K minus one, the control input from one time sample before
‫we were treated as an additional state and then Delta at K would be an additional state one time sample
‫later.
‫So this thing here would be actually part of our present state vector and this would be part of our
‫future state vector at K plus one.
‫So the state vector, it looked like this in the previous course, so why not at K K side on that K
‫and then the big inertial Y at K, but then we also added our Delta at K minus one here, and this vector
‫became an augmented state vector and augmented state vector.
‫And the notation that we use for an augment a state vector was that we added a tilde here and then the
‫state vector at K plus one would equal like this the same states but at K plus one.
‫But then we added the delta at K here and it became an augmented future state vector at K plus one.
‫And so to do that, to augment our state vectors, to expand them, we have to augment or expand our
‫system itself to incorporate this and this into our state's base equations.
‫And so the same thing will happen in the ugly case as well, instead of.
‫The control input use of key plus I, our control input will be dealt the vector at K plus I, or you
‫can write it out like this as well.
‫Delta you to add K plus I Delta U three at K plus I and Delta U for at K plus I.
‫And so we will define our delta.
‫You at K like this, the U vector at K minus the U vector at cave minus one.
‫We can then rewrite it like this you vector at K which is this one here equals U vector at K minus one
‫which is this one.
‫Plus, Delta, you, Victor and Kay, which is this one here, so our discrete states, base equations
‫can be rewritten like this, your state vector at Cape plus one equals a matrix times, your state vector
‫at K plus B times your regional control input vector atk.
‫But then you can rewrite the same thing like this where instead of use up k, you write it out in this
‫form like it is over here you add K minus one plus Delta U at K like this.
‫So this thing in the brackets that would be your use up K and of course you can open the brackets so
‫X and K plus one equals eight times X and K plus B times you add K minus one and plus B times Delta
‫you add K.
‫Then I'm also going to take this equation and I'm going to write it here as well.
‫You at K equals you at K minus one, plus Delta, you add K and this is a system of equations and now
‫you can augment your system like this.
‫This is your ex and Kate plus one, this is your you at K, you pack it in in one vector.
‫Then you will have a matrix here.
‫And you will have a vector here, so here you will have X and K, U at K minus one and then you will
‫have another Matrix here times Delta, you at K.
‫So this belongs to this, then this belongs to this.
‫Then this here belongs to this and this belongs to this, and also you have your you at K minus one
‫here, so that also belongs to this place here.
‫And then Delta you at K in both equations, they belonged to this one here.
‫And don't forget, you also had an output vector Y at K and that equals some kind of matrix and then
‫the augmented state vector.
‫So X at K would be here and then you add K minus one would be here and then you had another Matrix here
‫times Delta, you and K and now this is your exercice fill in this matrix here, this matrix here,
‫this matrix here and this matrix here.
‫What elements.
‫Those matrices would have tried yourself and then will continue in the next to you.
‫Thank you very much.