‫Now, we discussed that there were several NPC techniques available, such as basic NPC for linear time
‫invariant systems, and we used the abbreviation LTI, then robust NPC and non-linear NPC.
‫And what we wanted to do, we wanted to use the first one because it was the easiest to implement in
‫the system.
‫However, this system here that we had, we couldn't use our basic NPC for that because this formulation
‫here wasn't in the LTI form.
‫And so to achieve that LTI form, we made a small your angle approximation because we discussed that
‫if we go from one lane to another at, let's say, 20 meters per second, let's say like that, then
‫our your angle would be very small.
‫And we could just assume the following this final row here instead of writing it like this.
‫Thanks to the small you angle approximation, we could rewrite it like this big white dot equals small
‫white dot plus small x dot and pesi.
‫That allowed us to rewrite our state based system in the following form.
‫This was the system for the states and then we wanted to extract certain states.
‫And so we had another equation and that equation was for the outputs of the system.
‫And the outputs, at least in our case, were only this states that we had to compare to the reference
‫values.
‫And then what we did, we defined for matrices, Matrix A, Matrix B, Matrix C, and Matrix D, and
‫these are common matrices in the states based equations.
‫And because of the smaller, you wangel assumption, we were able to have these matrices, these four
‫matrices constant.
‫And when these four matrices are constant, then our system is in the LTI form and we can use the basic
‫NPC strategy and this entire thing here, these two systems of equations could be packaged in this generic
‫form.
‫And remember the state vector X, the time derivative of the state vector X dot the system input vector
‫you and the output vector Y, they are all a function of time around.
‫You can see it here.
‫They are all a function of time.