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‫Welcome back.

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‫I think that the best way to do this section is by briefly looking at what we had previously done in

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‫implementing the NPC controller to the vehicle.

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‫And then in parallel to that, we will look at how to implement these steps in the other case.

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‫That way you will see the similarities and the differences that would appear in this process of implementing

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‫the NPC controller to the UAE.

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‫So in case of the vehicle, we followed a similar process.

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‫We first hand our equations of motion in the lateral direction.

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‫Then we transformed them into the state's base equations, those states base equations were used in

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‫the plant of the vehicle.

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‫However, when it came to the NPC controller, we had to put these equations in the.

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‫Continues LTI system for.

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‫Or continuous linear time invariant form this LTI form of the state base equations was a special state

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‫space form in which you first put your state's space equations in this form.

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‫This one here where your state of vector, the control input vector, the output vector and then the

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‫time derivative of your state vector.

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‫They are all as a function of time.

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‫And then you made the assumptions possible in the systems states based equations.

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‫In order to make sure that the Matrixes, A, B, C and D will be completely constant as a function

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‫of time, you did not want these mattresses to change at all.

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‫And so when you're A, B, C and D matrices are completely constant in terms of time, then your system

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‫is in the LTI state space form.

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‫And once you have your system in the continuous LTI form, you have to democratize it.

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‫You would get your discrete LTI system.

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‫The main difference between the continuous and discrete system was the following.

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‫This was the continuous form here and this would be the discrete form here.

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‫The continuous LTI system gives you the time, derivatives of your status as a function of time, and

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‫then the discreetness system gives you a new state at K plus one.

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‫All right.

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‫So when you have a continuous form, you get a time derivative of your state as a function of time.

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‫And when you have a discrete form, then you get a new state, another time derivative of the state,

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‫but a new state at K plus one at one time, sample later.

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‫Because remember, now you don't have continuous time, you have time samples here and also that this

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‫critize states based system has different matrices.

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‫You have A sub D, B, sub DC, sub D and disturbed.

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‫In the previous course, I showed you how to derive the equations for these discrete matrices using

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‫the forward oilor method.

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‫The formulas that we have got were the following aid equals the identity matrix, plus the continuous

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‫a matrix times the time sample interval.

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‫Then BD was the continuous B matrix times the time sample interval.

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‫Then KDDI was equal to continue C and the D was equal to continues D.

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‫And then after that step.

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‫We use our discrete A, B, C and D matrices in the NPC control technique.

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‫Now, the DBE Matrix, and then they did Matrix, they were Xeros.

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‫Because remember the vector Y in the continuous form and also in the discrete form, it was our output

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‫vector and in our case and in most cases, the outputs are simply the states chosen in order to compare

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‫them with reference values.

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‫You would only need a D matrix if your output was some kind of mix between the states and the control

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‫inputs.

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‫For example, let's say that this is your state vector.

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‫You have one element X1 and then you have X2.

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‫And let's say that it's transposed because it's a common vector and let's say that you choose X one

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‫to be your output and then your output, which is X1 would be a matrix, which in this case would just

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‫be a role vector one and zero, and then you would multiply by the state vector x1 and X to.

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‫And this would be your see matrix, of course.

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‫And so this SI matrix would extract X one from this state vector, and that would become your output.

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‫And then obviously, you don't need anything from here, so your deal would be zero.

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‫You don't need to extract anything from the control input vector here, but then let's say that again,

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‫your state vector is X1 and X2.

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‫You have two states and then you control input vector is you one and you two, which is also common

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‫vector.

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‫But let's say that you want your output to be a mix of your state and control inputs and let's call

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‫it X and asterisk, then, for example, you could have something like this.

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‫You have to here you have five here.

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‫This would be your C matrix, of course.

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‫And then you multiply by your state vector and plus then you have something else here, one and two,

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‫so that would be your D matrix and you multiply it by you one and you two, which are your control inputs

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‫and then your output X asterisk would be two times X one plus five times X two plus one times you one

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‫and plus two times you two.

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‫So you see now your output is some kind of mix of your X one X two states and then you want and you

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‫to control inputs.

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‫But this kind of thing almost never happens in control engineering.

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‫In most cases, the outputs are simply states that you extract from the state vector because you have

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‫to compare that state against your reference value.

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‫X1 are.

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‫And therefore, in most cases, your democracies are zero.