# Nonlinear Robust Control of a Quadratic Boost Converter in a Wide Operation Range, Based on Extended Linearization Method

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Dynamic Model of the Quadratic Boost Converter

## 3. Controller Design

#### 3.1. Inner Control Loop with Sliding Mode Controller

#### 3.2. Dynamics of the Affine Model around the Desired Equilibrium State

#### 3.3. Structure of the Sliding Hyperplane for the Linearized Model (12)

**Remark**

**1.**

#### 3.4. Computation of the Sliding Function for the Switching Model (2)

**Remark**

**2.**

#### 3.5. Controller Scheme and Design of the Outer Control Loop

## 4. Experimental Set-Up

## 5. Experimental Results

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## References

- Carrasco, J.M.; Franquelo, L.G.; Bialasiewicz, J.T.; Galván, E.; PortilloGuisado, R.C.; Prats, M.M.; León, J.I.; Moreno-Alfonso, N. Power-electronic systems for the grid integration of renewable energy sources: A survey. IEEE Trans. Ind. Electron.
**2006**, 53, 1002–1016. [Google Scholar] [CrossRef] - Chen, J.; MaksimoviC, D.; Erickson, R. Buck-boost PWM converters having two independently controlled switches. In Proceedings of the Annual Power Electronics Specialists Conference, Vancouver, BC, Canada, 17–21 June 2001; IEEE: Piscataway, NJ, USA, 2001; Volume 2, pp. 736–741. [Google Scholar]
- Baek, J.W.; Ryoo, M.H.; Kim, T.J.; Yoo, D.W.; Kim, J.S. High boost converter using voltage multiplier. In Proceedings of the Annual Conference of IEEE Industrial Electronics Society, Raleigh, NC, USA, 6–10 November 2005; IEEE: Piscataway, NJ, USA, 2005; pp. 567–572. [Google Scholar]
- Navarro-López, E.M.; Cortés, D.; Castro, C. Design of practical sliding-mode controllers with constant switching frequency for power converters. Electr. Power Syst. Res.
**2009**, 79, 796–802. [Google Scholar] [CrossRef] - Oucheriah, S.; Guo, L. PWM-based adaptive sliding-mode control for boost DC–DC converters. IEEE Trans. Ind. Electron.
**2012**, 60, 3291–3294. [Google Scholar] [CrossRef] - Vidal-Idiarte, E.; Carrejo, C.E.; Calvente, J.; Martínez-Salamero, L. Two-loop digital sliding mode control of DC–DC power converters based on predictive interpolation. IEEE Trans. Ind. Electron.
**2010**, 58, 2491–2501. [Google Scholar] [CrossRef] - Wai, R.J.; Shih, L.C. Design of voltage tracking control for DC–DC boost converter via total sliding-mode technique. IEEE Trans. Ind. Electron.
**2010**, 58, 2502–2511. [Google Scholar] [CrossRef] - Deaecto, G.S.; Geromel, J.C.; Garcia, F.S.; Pomilio, J.A. Switched affine systems control design with application to DC–DC converters. IET Control Theory Appl.
**2010**, 4, 1201–1210. [Google Scholar] [CrossRef] - Baumann, W.; Rugh, W. Feedback control of nonlinear systems by extended linearization. IEEE Trans. Autom. Control
**1986**, 31, 40–46. [Google Scholar] [CrossRef] - Sira-Ramirez, H.; Rios-Bolívar, M. Synthesis of sliding-mode controllers for nonlinear systems via extended linearization. Dyn. Control
**1991**, 1, 379–403. [Google Scholar] [CrossRef] - Sferlazza, A.; Martínez-Salamero, L.; Sanchez, C.A.; Garcia, G.; Alonso, C. Min-Type Control Strategy of a DC–DC Synchronous Boost Converter. IEEE Trans. Ind. Electron.
**2019**, 67, 3167–3179. [Google Scholar] [CrossRef] [Green Version] - Ortiz-Lopez, M.; Leyva-Ramos, J.; Diaz-Saldierna, L.; Garcia-Ibarra, J.; Carbajal-Gutierrez, E. Current-mode control for a quadratic boost converter with a single switch. In Proceedings of the IEEE Power Electronics Specialists Conference, Orlando, FL, USA, 17–21 June 2007; IEEE: Piscataway, NJ, USA, 2007; pp. 2652–2657. [Google Scholar]
- Lopez-Santos, O.; Martinez-Salamero, L.; Garcia, G.; Valderrama-Blavi, H.; Sierra-Polanco, T. Robust sliding-mode control design for a voltage regulated quadratic boost converter. IEEE Trans. Power Electron.
**2014**, 30, 2313–2327. [Google Scholar] [CrossRef] - Barreto, L.H.S.C.; Coelho, E.A.A.; Farias, V.J.; de Oliveira, J.C.; de Freitas, L.C.; Vieira, J.J.B. A quasi-resonant quadratic boost converter using a single resonant network. IEEE Trans. Ind. Electron.
**2005**, 52, 552–557. [Google Scholar] [CrossRef] - Sferlazza, A.; Albea-Sanchez, C.; Garcia, G. A hybrid control strategy for quadratic boost converters with inductor currents estimation. Control Eng. Pract.
**2020**, 103, 104602. [Google Scholar] [CrossRef] - Albea, C.; Sferlazza, A.; Gordillo, F.; Gomez-Estern, F. Control of Power Converters With Hybrid Affine Models and Pulse-Width Modulated Inputs. IEEE Trans. Circuits Syst. I Regul. Pap.
**2021**, 68, 3485–3494. [Google Scholar] [CrossRef] - Sira-Ramirez, H.; Rios-Bolivar, M. Sliding mode control of DC-to-DC power converters via extended linearization. IEEE Trans. Circuits Syst. I Fundam. Theory Appl.
**1994**, 41, 652–661. [Google Scholar] [CrossRef] - Shtessel, Y.B.; Zinober, A.S.; Shkolnikov, I.A. Sliding mode control of boost and buck-boost power converters using method of stable system centre. Automatica
**2003**, 39, 1061–1067. [Google Scholar] [CrossRef] - Shtessel, Y.B.; Zinober, A.S.I.; Shkolnikov, I.A. Sliding mode control of boost and buck-boost power converters using the dynamic sliding manifold. Int. J. Robust Nonlinear Control
**2003**, 13, 1285–1298. [Google Scholar] [CrossRef] - Sanchez, C.A.; Garcia, G.; Sabrina, H.; Heemels, W.; Zaccarian, L. Practical stabilization of switched affine systems with dwell-time guarantees. IEEE Trans. Autom. Control
**2019**, 64, 4811–4817. [Google Scholar] [CrossRef] [Green Version] - Liu, J.; Laghrouche, S.; Harmouche, M.; Wack, M. Adaptive-gain second-order sliding mode observer design for switching power converters. Control Eng. Pract.
**2014**, 30, 124–131. [Google Scholar] [CrossRef] [Green Version] - DeCarlo, R.A.; Zak, S.H.; Matthews, G.P. Variable structure control of nonlinear multivariable systems: A tutorial. Proc. IEEE
**1988**, 76, 212–232. [Google Scholar] [CrossRef] [Green Version] - Utkin, V. Variable structure systems with sliding modes. IEEE Trans. Autom. Control
**1977**, 22, 212–222. [Google Scholar] [CrossRef] - Sira-Ramírez, H. Sliding Mode Control: The Delta-Sigma Modulation Approach; Birkhäuser: Basel, Switzerland, 2015. [Google Scholar]
- Messina, A.A.; Imbruglia, A.; Calabretta, M.; Vinciguerra, V.; Moise, C.C.; Sitta, A.; Enachescu, M.; Roccaforte, F. The “first and euRopEAn siC eighT Inches pilOt liNe”: A project, called REACTION, that will boost key SiC Technologies upgrading (developments) in Europe, unleashing Applications in the Automotive Power Electronics Sector. In Proceedings of the International Conference of Electrical and Electronic Technologies for Automotive, Turin, Italy, 18–20 November 2020; IEEE: Piscataway, NJ, USA, 2020; pp. 1–6. [Google Scholar]

**Figure 12.**${V}_{C1}$ and ${V}_{out}$ voltages, during a load variation from ${R}_{0}=380\phantom{\rule{4pt}{0ex}}\mathsf{\Omega}$ to ${R}_{0}=220\phantom{\rule{4pt}{0ex}}\mathsf{\Omega}$, with integral action.

**Figure 13.**${I}_{L1}$ and ${I}_{L2}$ currents, during a load variation from ${R}_{0}=380\phantom{\rule{4pt}{0ex}}\mathsf{\Omega}$ to ${R}_{0}=220\phantom{\rule{4pt}{0ex}}\mathsf{\Omega}$, with integral action.

**Figure 14.**${V}_{C1}$ and ${V}_{out}$ voltages, during a supply voltage variation from ${V}_{IN}=24$ V to ${V}_{IN}=20$ V, with integral action.

**Figure 15.**${I}_{L1}$ and ${I}_{L2}$ currents, during a supply voltage variation from ${V}_{IN}=24$ V to ${V}_{IN}=20$ V, with integral action.

Component | Value | Model | Description |
---|---|---|---|

${V}_{IN}$ | 24 V | Input Voltage | |

${L}_{1}$ | 330 $\mathsf{\mu}$H | AGP4233-334ME | Inductor |

${L}_{2}$ | 470 $\mathsf{\mu}$H | AGP4233-474ME | Inductor |

${r}_{{L}_{1}}$, ${r}_{{L}_{1}}$ | 11.5 m$\mathsf{\Omega}$ | Equivalent series resistance of the inductors | |

${C}_{1}$, ${C}_{2}$ | 20 $\mathsf{\mu}$F | MKP1848C62090JP4 | Capacitors |

${r}_{{C}_{1}}$, ${r}_{{C}_{1}}$ | 5 m$\mathsf{\Omega}$ | Equivalent series resistance of the capacitor | |

${R}_{0}$ | 380 $\mathsf{\Omega}$ | Load Resistor | |

${D}_{1,2,3}$ | C3D06060A | Diodes | |

${S}_{1}$ | C3M0065090D | Switch | |

Driver | 1EDI20N12AF | Switch Driver |

Settling Time | Overshoot | |||||||
---|---|---|---|---|---|---|---|---|

Test | Current | Voltage | Current | Voltage | ||||

${\mathit{I}}_{\mathit{L}\mathbf{1}}$ | ${\mathit{I}}_{\mathit{L}\mathbf{2}}$ | ${\mathit{V}}_{\mathit{C}\mathbf{1}}$ | ${\mathit{V}}_{\mathit{OUT}}$ | ${\mathit{I}}_{\mathit{L}\mathbf{1}}$ | ${\mathit{I}}_{\mathit{L}\mathbf{2}}$ | ${\mathit{V}}_{\mathit{C}\mathbf{1}}$ | ${\mathit{V}}_{\mathit{OUT}}$ | |

Start up | 2.5 ms | 2.1 ms | 2 ms | 2.2 ms | 3 A | 1.6 A | 0 V | 0 V |

Load Variation | 30 ms | 5 ms | 25 ms | 40 ms | 0 A | 0 A | 3.5 V | 9 V |

input Variation | 14 ms | 11 ms | 14 ms | 13 ms | 1.3 A | 0.5 A | 12 V | 6 V |

Start-up with PI | 4.3 ms | 3.9 ms | 3.8 ms | 4 ms | 18 A | 3.6 A | 4.5 V | 9 V |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Alonge, F.; Busacca, A.; Calabretta, M.; D’Ippolito, F.; Fagiolini, A.; Garraffa, G.; Messina, A.A.; Sferlazza, A.; Stivala, S.
Nonlinear Robust Control of a Quadratic Boost Converter in a Wide Operation Range, Based on Extended Linearization Method. *Electronics* **2022**, *11*, 2336.
https://doi.org/10.3390/electronics11152336

**AMA Style**

Alonge F, Busacca A, Calabretta M, D’Ippolito F, Fagiolini A, Garraffa G, Messina AA, Sferlazza A, Stivala S.
Nonlinear Robust Control of a Quadratic Boost Converter in a Wide Operation Range, Based on Extended Linearization Method. *Electronics*. 2022; 11(15):2336.
https://doi.org/10.3390/electronics11152336

**Chicago/Turabian Style**

Alonge, Francesco, Alessandro Busacca, Michele Calabretta, Filippo D’Ippolito, Adriano Fagiolini, Giovanni Garraffa, Angelo Alberto Messina, Antonino Sferlazza, and Salvatore Stivala.
2022. "Nonlinear Robust Control of a Quadratic Boost Converter in a Wide Operation Range, Based on Extended Linearization Method" *Electronics* 11, no. 15: 2336.
https://doi.org/10.3390/electronics11152336