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‫Welcome back.

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‫So in the previous course, I explained to you that this cost function form is a very good one because

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‫you only have one independent variable there, and that is Delta, you vector, and this is what you're

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‫after.

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‫You want to find the optimum delta, you global that looks like this Delta U at K, Delta U at Cape

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‫plus one Delta you K plus two and Delta you at Cape plus three and transposed.

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‫So you want the NPC to find that so that you could take this first element here and add it to the previous

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‫you.

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‫So that was your previous you and then you add this delta, you add key to the previous you and you

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‫will get your you at K and this is what goes into the plant.

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‫Alternatively, you can expand this expression and write it out like this.

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‫And again, this one will go into your plant like this, this vector here and the other Delta use here

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‫will they will be neglected.

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‫And so this cost function form is very good because you don't have your future augmented state vectors

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‫like X till the global.

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‫You don't have it as a variable here in this cost function.

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‫For since your double prime does not have those global state vectors as an independent variable, you

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‫only have this X still that K transposed, which is your present known state and therefore a constant.

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‫But since there is no X still the global that contains future state vectors, augmented ones finding

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‫this delta you global vector is super easy.

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‫You simply take the gradient of g the prime that looks like this h double ba delta u g plus f double

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‫ba times this vector your present augmented state vector and then your global reference vector.

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‫So that's your gradient and then you make it equal to zero.

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‫In the previous course I explained extensively why the gradient is like that.

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‫So since it is a positive quadratic function, when you make the gradient of G double primes zero,

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‫it allows you to find values for Delta U global that would minimize this G double prime.

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‫And if you remember, that was the whole point, finding the Delta U global values that minimize your

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‫cost function, because through it you will find control inputs that will minimize your error while

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‫also keeping the change of your control actions as small as possible.

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‫That's why your original cost function started with two types of variables the error variables and then

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‫the delta you variables.

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‫You want to minimize your errors, but you don't want your you to use three and you for change like

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‫crazy.

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‫And so finding this delta, you global is very easy.

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‫You have this h double ba delta, you global and then the other term you put it on the other side minus

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‫F double ba.

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‫Your present augment is state vector and then your global reference vector like this.

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‫And then what you do you want to eliminate this double bar here so that you would only be left with

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‫this one here, which is this one here.

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‫So for that you're going to put h a double bar inverse here and also you have to put it here h double

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‫bar inverse.

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‫Then this thing here will become an identity matrix or one in the scale of world.

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‫And so your delta, you global equals minus H double bar inverse times F double bar times this vector

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‫and there you go.

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‫You have your formula for your delta you globe.

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‫Now, there is one important distinction that you have to make this part here.

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‫It was constant in the autonomous vehicle case.

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‫If you look at the entire cost function derivation, then H double bar and F double bar matrices ultimately

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‫contain discrete Amy ptosis in them.

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‫And in the previous course we really had an LTI system, meaning that our discrete A never changed throughout

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‫the entire maneuver.

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‫And this is why your H double bar and then your F double bar or in this case F double bar transposed.

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‫This is why they always remained unchanged.

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‫That's not the case with the US, though.

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‫Every time you send you to Q3 and Q4 to the plant, the plant will give you new states and it will also

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‫give you a new Omega.

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‫So your new P, Q R states will produce new PHY that to that and say that after you put them through

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‫the transfer matrix and then your new FY that theta that and Omega will change your continues APV a

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‫matrix and then after you democratize it you will have a different discrete a matrix, you know a different

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‫a sub D and that means that you're H double bar and F double bar transposed.

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‫And because of that your H double bar inverse and your F double bar, that means that they will also

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‫change.

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‫Remember, we don't have an LTI system here.

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‫We have a nonlinear system, but we used a linear parameter varying or Lopevi approach.

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‫Essentially we squeezed a nonlinear system into a linear system format.

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‫And inside the A matrix, we kept some of its elements as variables that we would update.

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‫Every D equals zero point one second.

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‫Every time we had an inner loop, we updated our A matrix and for smooth trajectories that we have,

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‫this approach works very well.

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‫HPV is a highly researched topic in robust control.

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‫Their entire books written about it.

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‫One book that I know is this one robust control and linear parameter varying approach by Ollivier cinema.

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‫Patrick Gasparini was Chef Bocker and that's it for this section.

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‫You have seen that there are many similarities in using NPC in the autonomous vehicle, lateral control

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‫and the use of adequate control.

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‫Of course, you have to make some adaptations, but at its core, the logic of the technique is the

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‫same.

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‫NPC is a very powerful control technique.

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‫In the next section, we will take a look at another nonlinear control technique called feedback polymerization,

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‫and that will be for the position control.

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‫Thank you very much.

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‫And I see you in the next section.