# Control of a DC-DC Buck Converter through Contraction Techniques

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Methods

#### 2.1. Linear Transformations

#### 2.2. Matrix Measure

#### 2.3. Contraction Analysis for Filippov Systems

## 3. The Buck Power Converter

## 4. Application to 2D-Case

#### 4.1. Controller Design

#### 4.2. Simulation Results

## 5. Application to 3D-Case

#### 5.1. Controller Design Based on a Modified Integral Control Action

#### Simulation Results

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Matrix Measure for a 2D System

**First order principal minors**. The matrix has two first order principal minors which are:

**Second order principal minors**. This system has only one second order principal minor which is computed as:

#### Appendix A.2. Matrix Measure for 3D System

**First order principal minors**. Here, there are three first order principal minors, they are:

**Second order principal minor**. In this case, there are three second order principal minors. The first one is:

**Third order principal minor**. As matrix N is obtained from two vectors, its range cannot be greater than two, then its third order principal minor namely

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**Figure 2.**(

**A**) Time trace of the voltage in the capacitor v. During the first 30 ms a ${\overline{v}}_{ref}=32$ V is used, after this a drastic change to ${\overline{v}}_{ref}=16$ V is applied (depicted in the dashed lines). The time trace of the steady state percentage error is also depicted in the insets for both values of ${\overline{v}}_{ref}$; (

**B**) Phase representation of the steady state for ${\overline{v}}_{ref}=32$; (

**C**) ${\overline{v}}_{ref}=16$, with the equilibrium point indicated by the red star. Simulations were performed using MATLAB

^{®}with a fourth order Runge-Kutta algorithm with variable step and event detection to identify collisions with the hysteresis band. Steady state was considered after 20 ms of simulation time. Initial conditions were chosen as $(v,i)=(0,0)$. Other parameters as in the main text.

**Figure 3.**Time response of the capacitor’s voltage v. During the first 30 ms, the value of the resistor is set to $R=20\phantom{\rule{0.166667em}{0ex}}\mathsf{\Omega}$, after this the load is changed to $R=18\phantom{\rule{0.166667em}{0ex}}\mathsf{\Omega}$. Inset: Steady state percentage error (considered 15 ms after the presentation of the disturbance). The desired output is plot with the dashed line. Other details as in Figure 2.

**Figure 4.**(

**a**) Time trace of the voltage in the capacitor v. During the first 40 ms a ${\overline{v}}_{ref}=32$ V is used, after this a drastic change to ${\overline{v}}_{ref}=16$ V is applied (depicted in the dashed lines). The time trace of the steady state percentage error is also depicted in the insets for both values of ${\overline{v}}_{ref}$; (

**b**) Phase representation of the steady state for ${\overline{v}}_{ref}=32$; (

**c**) ${\overline{v}}_{ref}=16$, with the equilibrium point indicated by the red star. This results were obtained by making ${c}_{1}/{c}_{2}=9$ and $\delta =1\times {10}^{-4}$. Steady state was considered after 25 ms of transient dynamics. Other parameters as in the main text and Figure 2.

**Figure 5.**(

**a**) Time response of the capacitor’s voltage v. During the first 40ms, the value of the resistor is set to $R=20\phantom{\rule{0.166667em}{0ex}}\mathsf{\Omega}$, after this the load is changed to $R=15\phantom{\rule{0.166667em}{0ex}}\mathsf{\Omega}$. Inset: Steady state percentage error (considered 25 ms after the presentation of the disturbance). The desired output is plot with the dashed line; (

**b**) Same as (

**A**) for a disturbance in the input voltage E. During the first 40 ms $E=40$ V, after this it is changed to $E=50$ V. In this panel, the inset shows the percentage error during the first 20 ms after the presentation of the input disturbance. Other details as in Figure 2.

**Figure 6.**(

**A**) Black symbols: Settling time of the buck converter with different values of $\sqrt{LC}$ and fixed $R=20\phantom{\rule{0.166667em}{0ex}}\mathsf{\Omega}$, preserving the value of $\gamma =0.3536$. Red line: Linear fit of the simulation points. For this plot, we chose inductance values in the range $L=[20\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}$H 10 mH] and a capacitance $C=L/50$ F. Settling time was calculated as the time it takes to the system to evolve from $(v,\phantom{\rule{0.277778em}{0ex}}i,\phantom{\rule{0.277778em}{0ex}}y)=(0,0,0)$ to the point in which the error doesn’t leave the $\pm 2\%$ band. Inset: Switching frequency resulting from the hysteresis band in Equation (33) for the different values of $\sqrt{LC}$ reported in the main figure (black symbols). Red line depicts the fitted function reported in the text box; (

**B**) Average steady state error calculated as the mean value of the error after the settling time; (

**C**) percentage overshoot for the values of $\sqrt{LC}$ reported in panel (

**A**). For each simulation point, the system is evolved during a time span of $T=200\sqrt{LC}$. Other details as in Figure 2.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Angulo-Garcia, D.; Angulo, F.; Osorio, G.; Olivar, G.
Control of a DC-DC Buck Converter through Contraction Techniques. *Energies* **2018**, *11*, 3086.
https://doi.org/10.3390/en11113086

**AMA Style**

Angulo-Garcia D, Angulo F, Osorio G, Olivar G.
Control of a DC-DC Buck Converter through Contraction Techniques. *Energies*. 2018; 11(11):3086.
https://doi.org/10.3390/en11113086

**Chicago/Turabian Style**

Angulo-Garcia, David, Fabiola Angulo, Gustavo Osorio, and Gerard Olivar.
2018. "Control of a DC-DC Buck Converter through Contraction Techniques" *Energies* 11, no. 11: 3086.
https://doi.org/10.3390/en11113086