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‫Welcome back.

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‫So let's now shift our attention to the drone and let's see how to apply constraints to it.

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‫I mentioned that the drone rotors have a minimum and the maximum rotational speed.

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‫So the Omegas of your rotors had to stay between these minimum and maximum values.

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‫You now know that when you want to apply constraints to your embassy controller, then you have to use

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‫solvers for that in order to properly use solvers.

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‫You have to give it your bull bar matrix and small if transposed vector, which come from your cost

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‫function and also your matrix and each vector that come from your constraints.

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‫That way, the solver knows what kind of cost, function and constraints you have.

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‫The key for knowing exactly what constraints we are talking about is that the solver assumed that your

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‫constraints are in this form.

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‫G times delta, you global less than or equal to.

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‫And then this h vector.

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‫It's extremely important that your inequality sign points to the left.

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‫Now what is Delta you global in our case?

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‫Well, if Horizon period equals four, then your delta you global equals all that.

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‫First of all, you have Delta use at K, then you have them at K plus one.

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‫Then you have them as K plus two.

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‫And then you have them at K Plus three, you have 12 elements in this vector.

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‫Now, if we wanted to constrain our delta use directly, then it would have been quite easy.

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‫Let's say that you have something like this your delta, your global vector, which is this one here.

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‫And remember, it's transposed.

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‫So actually as a vector, it's between these minimum values and these maximum values here.

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‫So you simply give some kind of meaning values to all these elements and then some kind of maximum values

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‫to all these elements.

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‫And they will be in this vector here and this vector here, respectively.

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‫And now you can separate them like this.

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‫You can take this part.

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‫Let's call one.

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‫And then you can take this part here, and let's call it two.

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‫And so this would be the first part, and this would be the second part.

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‫Now, remember, in order to use the solver, all your inequality signs must point to the left.

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‫And so in the second part, you have to multiply this one by minus one.

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‫And this one by minus one and then your inequality sine changes its direction like this.

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‫But I can also rewrite them.

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‫Like this, I can have here an identity matrix, which is twelve by twelve times Delta U.

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‫Global, less than or equal to Delta U.

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‫Global Max and then minus an identity matrix.

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‫Twelve by twelve times Delta you global less than or equal to minus Delta you global mean or I can rewrite

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‫all that like this.

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‫I have a twelve by twelve identity matrix here and then minus and then the same identity matrix times

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‫delta you global less than or equal to.

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‫And then I have Delta U.

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‫Global Max here and minus Delta U.

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‫Global mean.

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‫And this i twelve by twelve.

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‫That's simply your identity matrix.

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‫It's simply your twelve by twelve identity matrix.

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‫When you multiply by Delta, you global, then it's like multiplying something by one in one dimension.

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‫And so that means that this matrix here, it has 24 rows and 12 columns.

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‫Then Delta You Global has twelve rows and one column, and then this vector here it has 24 rows and

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‫one column.

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‫So the dimensions work out.

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‫And then this matrix that your g matrix, this vector, that's your h vector, your inequality sine

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‫points in the right direction.

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‫And so that's what you would give to your solver.

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‫This matrix and h vector, together with your h double bar matrix and small f transpose to vector.

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‫And so you would simply write Delta your global.

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‫See, which means that you take constraints into account.

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‫It equals soul, underscore kewpie age, double bar small.

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‫I've transposed G h vector transposed and then you would right here solver equals CV x orbit.

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‫Or if you work in MATLAB, then you would write Delta you global.

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‫Considering the constraints equals what?

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‫Prog?

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‫And then again, the same each double bar matrix small if transposed the G matrix and the H vector transposed.

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‫And of course, in MATLAB, this one here and this one here, they can also be column vectors.

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‫And that's how you find a solution that minimizes your cost function.

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‫But make sure that you stay between your delta, your global men and delta, you global max, whatever

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‫those values are.

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‫However, that's not what we want in reality.

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‫In reality, we want to constrain our Omegas.

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‫So let's see how we are going to do it.

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‫Well, first of all, we have to define what our omega men and omega max values are.

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‫From what I've seen in other research papers for the UAV that we are considering, the reasonable values

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‫are omega men equals 110 times pi over three radians per second and omega max eight hundred and sixty

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‫pi over three radians per second.

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‫So approximately this one is 115 radians per second, and this one, it's 900 radians per second.

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‫That's great.

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‫So we have our omega minimum and omega maximum, but now we have to find a way to take that information

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‫and convert it into this form.

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‫Our familiar G times delta your global less than or equal to.

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‫And then the H vector.

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‫So how would we do it?

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‫Well, let me give you a thought exercise here before watching the next video.

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‫Give it a thought only maybe for five or ten minutes or so.

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‫And just think about all the possible ways you could go from your Omegas to your delta use and then

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‫how to eventually reach this form here.

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‫Just give it a thought, and let's continue in the next video.

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‫Thank you very much.