Power Electronics Handbook, 4th Edition

35.4.4 Examples: Predictive Control of Multilevel Inverters and Matrix Converters

35.4.4.1 Example 35.19: Non-linear Predictive Optimum Control of Multilevel Inverters

Consider the multilevel inverter of Fig. 35.58, with three-phase ac current direct dynamics in the αβ plane given by (35.167) and 35.161, which can be rewritten as

$\begin{array}{l}\left[\begin{array}{c}\frac{\mathrm{d}{i}_{\alpha }}{\mathrm{d}t}\\ \frac{\mathrm{d}{i}_{\beta }}{\mathrm{d}t}\\ \frac{\mathrm{d}{U}_{C1}}{\mathrm{d}t}\\ \frac{\mathrm{d}{U}_{C2}}{\mathrm{d}t}\end{array}\right]=\left[\begin{array}{cccc}-\frac{R}{L}& 0& \frac{{\Gamma }_{1\alpha }}{L}& \frac{{\Gamma }_{2\alpha }}{L}\\ 0& -\frac{R}{L}& \frac{{\Gamma }_{1\beta }}{L}& \frac{{\Gamma }_{2\beta }}{L}\\ -\frac{{\Gamma }_{1\alpha }}{C_1}& -\frac{{\Gamma }_{1\beta }}{C_1}& 0& 0\\ -\frac{{\Gamma }_{2\alpha }}{C_2}& -\frac{{\Gamma }_{2\beta }}{C_2}& 0& 0\end{array}\right]\left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\\ U_{C1}\\ U_{C2}\end{array}\right]+\left[\begin{array}{ccc}-\frac{1}{L}& 0& 0\\ 0& -\frac{1}{L}& 0\\ 0& 0& \frac{1}{C_1}\\ 0& 0& \frac{1}{C_2}\end{array}\right]\left[\begin{array}{c}{U}_{e\alpha }\\ U_{e\beta }\\ i_{\mathit{dc}}\end{array}\right].\end{array}$

  (35.241)

where ${\Gamma }_{i\alpha }=\sqrt{\frac{2}{3}}\left({\Gamma }_{i1}-\frac{{\Gamma }_{i2}}{2}-\frac{{\Gamma }_{i3}}{2}\right)$, ${\Gamma }_{i\beta }=\sqrt{\frac{2}{3}}\left(\frac{\sqrt{3}}{2}{\Gamma }_{i2}-\frac{\sqrt{3}}{2}{\Gamma }_{i3}\right)$, and i∈{1,2}.

The optimal controller is designed ...