‫And in order to use the basic NPC version, we first had to democratize our system because it was in
‫the continuous form.
‫But NPC is a discrete control technique.
‫One way to look at the difference between a continuous and a discrete system mathematically was to look
‫at their respective states based system forms.
‫So this one was for the continuous system and this one was for the discrete system where the discrete
‫AI was the identity matrix plus the continuous AI, which is this one times the sample time, then the
‫discrete B was the continuous B, which is this one times the sample time and then the discrete C was
‫the same, like the continuous C, which is this one, and the discrete D was the same, like the continuous
‫deal which is this one.
‫And we said that in most cases this part here becomes zero and this part here becomes zero as well.
‫And these expressions here, they come from a simple democratization technique called the forward oilor
‫method.
‫Note the difference in the logic in the continuous case after multiplying your states, which is this
‫one and your inputs, which is this one, after multiplying them by the matrices and then adding them
‫together, you get the time derivative of your states.
‫But in the discrete version, after doing the same thing, you get a state, an entire state in the
‫next sample time.
‫That's one way to look at their differences in the continuous form.
‫You get that time derivative of your states and the output as a function of time.
‫And in the district form you get an entire state in the next sample time and an output in this present
‫sample time.
‫So the state will be one sample time ahead compared to the output and the discrete system matrices.
‫Add BDC and are the ones that will be used in NPC.
‫A good explicit way to see why we needed to democratize the system can be seen.
‫If we look at this mathematical manipulation right here.
‫Consider this system with the horizon period of three samples.
‫The goal here is to lose all the future state variables on the right side of the equations X1 and x2
‫so that you would only be left with your current state X zero and your system inputs Delta Zero, Delta
‫one and Delta to this manipulation is just one part of the entire NPC derivation that I went through
‫thoroughly applied control systems for engineers one.
‫However, it shows very well that you need a discrete system form and not a continuous version because
‫on the left side of the equations you need to have the new states in the next sample type and not a
‫derivative of the states with respect to time.