# Comparison of Different Repetitive Control Architectures: Synthesis and Comparison. Application to VSI Converters

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## Abstract

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## 1. Introduction

## 2. Repetitive Control Basics and Architectures

#### 2.1. Periodical Signal Generator

#### 2.2. Series Approach

#### 2.3. Plug-in Approach

- ${T}_{o}\left(z\right)$ must be stable (${G}_{c}\left(z\right)$ can be designed to fulfill it)
- $\parallel W\left(z\right)H\left(z\right)\left(1-{k}_{r}\right){\parallel}_{\infty}<1$ (${k}_{r}$ can be selected appropriately)

#### 2.4. Disturbance Rejection Approach

#### 2.5. Repetitive Control Using the Stabilizing Controllers’ Parametrization

#### 2.6. ${H}_{\infty}$ Scheme

## 3. Repetitive Control Design for the Voltage Source Inverter

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 10.**Magnitude Bode diagram of ${W}_{u}=\frac{{G}_{d}\left(s\right)}{Z\left(s\right)+{G}_{d}\left(s\right)}$ for different load values: ${R}_{L}\in \left[10\phantom{\rule{0.277778em}{0ex}}100\right]\phantom{\rule{0.277778em}{0ex}}\Omega $ and ${L}_{L}\in \left[10\phantom{\rule{0.277778em}{0ex}}10000\right]\phantom{\rule{0.277778em}{0ex}}\mu H$, ${W}_{u}\left(s\right)$, and the frequency response of the bounding function ${W}_{u}^{m}\left(s\right)$.

**Figure 11.**Sensitivity function for RC based on the series, plug-in, disturbances observer, and Youla approaches.

**Figure 12.**Sensitivity function comparison for RC based on the series and ${H}_{\infty}$ approaches.

**Figure 16.**Experimental results for the nonlinear load. From top to bottom: series, plug-in, disturbance rejection, Youla, and robust designs. From left to right: voltage and load current wave forms, voltage harmonics, current harmonics, and power.

Harmonics | RC | HORC |
---|---|---|

Full | $\begin{array}{c}I\left(z\right)=\frac{H\left(z\right)}{{z}^{N}-H\left(z\right)}\\ W\left(z\right)={z}^{-N},\sigma =1\end{array}$ | $\begin{array}{c}I\left(z\right)=\frac{W\left(z\right)H\left(z\right)}{1-W\left(z\right)H\left(z\right)}\\ W\left(z\right)=1-{\left(1-{z}^{-N}\right)}^{M},\sigma =1\end{array}$ |

Odd | $\begin{array}{c}I\left(z\right)=\frac{-H\left(z\right)}{{z}^{\frac{N}{2}}+H\left(z\right)}\\ W\left(z\right)={z}^{-\frac{N}{2}},\sigma =-1\end{array}$ | $\begin{array}{c}I\left(z\right)=\frac{-W\left(z\right)H\left(z\right)}{1+W\left(z\right)H\left(z\right)}\\ W\left(z\right)=-1+{\left(1+{z}^{-\frac{N}{2}}\right)}^{M},\sigma =-1\end{array}$ |

$6l\pm 1$ [23] | $\begin{array}{c}I\left(z\right)=\frac{W\left(z\right)H\left(z\right)}{1+W\left(z\right)H\left(z\right)}\\ W\left(z\right)={z}^{-\frac{N}{3}}-{z}^{-\frac{N}{6}},\sigma =-1\end{array}$ | - |

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**MDPI and ACS Style**

Ramos, G.A.; Costa-Castelló, R. Comparison of Different Repetitive Control Architectures: Synthesis and Comparison. Application to VSI Converters. *Electronics* **2018**, *7*, 446.
https://doi.org/10.3390/electronics7120446

**AMA Style**

Ramos GA, Costa-Castelló R. Comparison of Different Repetitive Control Architectures: Synthesis and Comparison. Application to VSI Converters. *Electronics*. 2018; 7(12):446.
https://doi.org/10.3390/electronics7120446

**Chicago/Turabian Style**

Ramos, Germán A., and Ramon Costa-Castelló. 2018. "Comparison of Different Repetitive Control Architectures: Synthesis and Comparison. Application to VSI Converters" *Electronics* 7, no. 12: 446.
https://doi.org/10.3390/electronics7120446