4.7 Working Principle

In order to illustrate how the predictive control strategy works, a detailed example is shown in Figure 4.5 and Figure 4.6. Here, load currents i_α, i_β, and their references are shown for a complete period of the reference. Using the measurement i(k) and all switching states of the voltage vector v(k), the future currents i(k + 1) are estimated, i^p(k + 1).

Figure 4.4 Voltage vectors in the complex plane

Figure 4.5 Working principle: vectorial plot of the reference and predicted currents

In the vectorial plot, shown in Figure 4.5, it can be observed that vector V₂ takes the predicted current vector closest to the reference vector.

As shown in Figure 4.6, current corresponds to the predicted current if the voltage vector V₀ or V₇ is applied at time k. It can be seen in this figure that vectors V₂ and V₆ are the ones that minimize the error in the i_α current, and vectors V₂ and V₃ are the ones that minimize the error in the i_β current, so the voltage vector that minimizes the cost function g is V₂.

These figures illustrate the meaning of the cost function as a measure of error or distance between reference and predicted vectors. It is easy to view these errors and distances for the case of current control, but these plots become difficult or impossible to build for more complex cost functions.

From a numerical point of view, the selection of the optimum voltage vector is performed as presented in Figure 4.7. Each voltage vector generates a predicted current that gives a value of the cost function, as listed in the table. It can be observed that, for this example, vector V₂ produces the lowest value of the cost function g. Then, voltage vector V₂ is selected and applied in the inverter.

Figure 4.6 Working principle: reference and predicted currents