# Design of Nonlinear Backstepping Double-Integral Sliding Mode Controllers to Stabilize the DC-Bus Voltage for DC–DC Converters Feeding CPLs

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## Abstract

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## 1. Introduction

## 2. System Description and Its Dynamical Model

## 3. Proposed Controller Design Approach

- Transformation of the model into an exactly linearized system using an exact feedback linearization approach and making it suitable to apply the backstepping control approach;
- Elimination of the non-minimum phase problem of DDBCs by considering the total stored energy and its rate of change as two new state variables;
- Derivation of the control law using the proposed scheme that satisfies the desired tracking performance to track the DC-bus voltage; and
- Stability analysis of the whole system with the newly derived control input performed using the Lyapunov stability theory.

#### 3.1. Transformation of the Model into an Exactly Linearized System Using the Exact Feedback Linearization

**Step 1:**Selection of the output function

**Step 2:**Relative degree calculation

**Step 3:**Nonlinear coordinate transformation and exact linearization

#### 3.2. Controller Design

**Step 1:**For fulfilling the design objective, the first tracking error $\left({e}_{1}\right)$ can be defined as follows:

**Step 2:**The dynamic of ${e}_{2}$ can be written as follows:

## 4. Simulation Results

**Case I:**Controller performance investigation with variations in the reference power of the CPL;**Case II:**Controller performance investigation with variations in the reference voltage of the DC-bus; and**Case III:**Controller performance investigation with variations in the input voltage.

**Case I:**Controller performance investigation with variations in the reference power of the CPL

**Case II:**Controller performance investigation with the variations in the reference voltage of the DC-bus

**Case III:**Controller performance investigation with the variations in the input voltage

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

BDI-SMC | Nonlinear Backstepping Double Integral-Sliding Mode Controller |

BSC | Nonlinear Backstepping Controller |

CPL | Constant Power Load |

CCM | Continuous Conduction Mode |

DGSs | Distributed Generation Systems |

DDBC | DC–DC Boost Converter |

DCDNs | DC Distribution Networks |

ESMC | Existing Sliding Mode Controller |

FBLCs | Nonlinear Feedback Linearizing Controllers |

MPC | Model Predictive Controller |

PV | Solar Photovoltaic |

PECs | Power Electronic Converters |

PI | Proportional-Integral Controller |

PWM | Pulse Width Modulation |

RESs | Renewable Energy Sources |

SMC | Sliding Mode Controller |

VPSs | Vehicular Power Systems |

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Parameters | Description | Value |
---|---|---|

${V}_{in}$ | Supply voltage | 55 V |

${V}_{bus}$ | Main DC-bus voltage | 110 V |

${P}_{CPL}$ | Nominal power in constant power load | 2 kW |

${r}_{b}$ | Parasitic resistance of an inductor | 2 m$\mathsf{\Omega}$ |

${L}_{b}$ | Inductance of the converter | 5 mH |

${C}_{bus}$ | Capacitance of the main DC-bus | 6 mF |

${R}_{Load}$ | Resistive Load | ∞ |

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## Share and Cite

**MDPI and ACS Style**

Ghosh, S.K.; Roy, T.K.; Pramanik, M.A.H.; Mahmud, M.A.
Design of Nonlinear Backstepping Double-Integral Sliding Mode Controllers to Stabilize the DC-Bus Voltage for DC–DC Converters Feeding CPLs. *Energies* **2021**, *14*, 6753.
https://doi.org/10.3390/en14206753

**AMA Style**

Ghosh SK, Roy TK, Pramanik MAH, Mahmud MA.
Design of Nonlinear Backstepping Double-Integral Sliding Mode Controllers to Stabilize the DC-Bus Voltage for DC–DC Converters Feeding CPLs. *Energies*. 2021; 14(20):6753.
https://doi.org/10.3390/en14206753

**Chicago/Turabian Style**

Ghosh, Subarto Kumar, Tushar Kanti Roy, Md. Abu Hanif Pramanik, and Md. Apel Mahmud.
2021. "Design of Nonlinear Backstepping Double-Integral Sliding Mode Controllers to Stabilize the DC-Bus Voltage for DC–DC Converters Feeding CPLs" *Energies* 14, no. 20: 6753.
https://doi.org/10.3390/en14206753