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‫And in order to use the basic NPC version, we first had to democratize our system because it was in

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‫the continuous form.

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‫But NPC is a discrete control technique.

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‫One way to look at the difference between a continuous and a discrete system mathematically was to look

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‫at their respective states based system forms.

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‫So this one was for the continuous system and this one was for the discrete system where the discrete

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‫AI was the identity matrix plus the continuous AI, which is this one times the sample time, then the

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‫discrete B was the continuous B, which is this one times the sample time and then the discrete C was

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‫the same, like the continuous C, which is this one, and the discrete D was the same, like the continuous

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‫deal which is this one.

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‫And we said that in most cases this part here becomes zero and this part here becomes zero as well.

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‫And these expressions here, they come from a simple democratization technique called the forward oilor

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‫method.

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‫Note the difference in the logic in the continuous case after multiplying your states, which is this

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‫one and your inputs, which is this one, after multiplying them by the matrices and then adding them

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‫together, you get the time derivative of your states.

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‫But in the discrete version, after doing the same thing, you get a state, an entire state in the

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‫next sample time.

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‫That's one way to look at their differences in the continuous form.

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‫You get that time derivative of your states and the output as a function of time.

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‫And in the district form you get an entire state in the next sample time and an output in this present

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‫sample time.

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‫So the state will be one sample time ahead compared to the output and the discrete system matrices.

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‫Add BDC and are the ones that will be used in NPC.

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‫A good explicit way to see why we needed to democratize the system can be seen.

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‫If we look at this mathematical manipulation right here.

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‫Consider this system with the horizon period of three samples.

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‫The goal here is to lose all the future state variables on the right side of the equations X1 and x2

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‫so that you would only be left with your current state X zero and your system inputs Delta Zero, Delta

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‫one and Delta to this manipulation is just one part of the entire NPC derivation that I went through

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‫thoroughly applied control systems for engineers one.

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‫However, it shows very well that you need a discrete system form and not a continuous version because

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‫on the left side of the equations you need to have the new states in the next sample type and not a

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‫derivative of the states with respect to time.