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‫Now, we discussed that there were several NPC techniques available, such as basic NPC for linear time

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‫invariant systems, and we used the abbreviation LTI, then robust NPC and non-linear NPC.

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‫And what we wanted to do, we wanted to use the first one because it was the easiest to implement in

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‫the system.

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‫However, this system here that we had, we couldn't use our basic NPC for that because this formulation

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‫here wasn't in the LTI form.

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‫And so to achieve that LTI form, we made a small your angle approximation because we discussed that

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‫if we go from one lane to another at, let's say, 20 meters per second, let's say like that, then

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‫our your angle would be very small.

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‫And we could just assume the following this final row here instead of writing it like this.

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‫Thanks to the small you angle approximation, we could rewrite it like this big white dot equals small

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‫white dot plus small x dot and pesi.

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‫That allowed us to rewrite our state based system in the following form.

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‫This was the system for the states and then we wanted to extract certain states.

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‫And so we had another equation and that equation was for the outputs of the system.

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‫And the outputs, at least in our case, were only this states that we had to compare to the reference

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‫values.

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‫And then what we did, we defined for matrices, Matrix A, Matrix B, Matrix C, and Matrix D, and

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‫these are common matrices in the states based equations.

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‫And because of the smaller, you wangel assumption, we were able to have these matrices, these four

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‫matrices constant.

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‫And when these four matrices are constant, then our system is in the LTI form and we can use the basic

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‫NPC strategy and this entire thing here, these two systems of equations could be packaged in this generic

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‫form.

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‫And remember the state vector X, the time derivative of the state vector X dot the system input vector

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‫you and the output vector Y, they are all a function of time around.

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‫You can see it here.

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‫They are all a function of time.