# Design of Continuous Finite-Time Controller Based on Adaptive Tuning Approach for Disturbed Boost Converters

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## Abstract

**:**

## 1. Introduction

- A novel adaptive continuous finite-time controller is designed for disturbed boost converter systems in the presence of white noise and model uncertainty;
- A design method which guarantees the convergence and maintenance of the system state trajectories to a predefined neighbourhood of the origin in the finite time;
- Estimating the disturbance boundaries using an adaptive continuous scheme to adjust the controller adaptation gain;
- Eliminating the chattering phenomenon using a switching function based on a continuous adaptation gain.

## 2. DC–DC Boost Converter

**Lemma**

**1**

**.**Consider $x\in \u01b4\subset {R}^{n}$ and $\dot{x}=\u03f4\left(x\right),\u03f4:{R}^{n}\to {R}^{n}$ as a continuous system, in the region $\u01b4$ as the open neighborhood of origin and locally Lipschitz at $\u01b4\left\{0\right\}$ and $\u03f4\left(0\right)=0$. Moreover, assume that the following conditions hold, if there is a continuous Lyapunov function $V:\u01b4\to R$:

- (a)
- $V$is positive-definite;
- (b)
- time derivative of $V$at $\u01b4\left\{0\right\}$ is negative-definite;
- (c)
- There are positive real values $m$ and $0<a<1$, and a neighbourhood $\u019d\subset \u01b4$ of origin such that:

## 3. Control Design

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

## 4. Simulation Results

**Scenario**

**1.**

**Scenario**

**2.**

**Scenario**

**3.**

**Scenario**

**4.**

**Scenario**

**5.**

**Scenario**

**6.**

**Scenario**

**7.**

**Scenario**

**8.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${S}_{\omega}$ | Switch | $\stackrel{~}{x}\left(t\right)$ | Reference signal |

$D$ | Fast diode | $e\left(t\right)$ | Error signal |

$R$ | Resistance | $\tau \left(t\right)$ | Sliding surface |

$L$ | Inductor | $\rho $ | Positive and constant parameter |

$C$ | Capacitor | $\eta $ | Positive parameter |

$u$ | Control input | $\sigma $ | Positive parameter |

${\dot{I}}_{L}$ | Inductor current | $z$ | Positive odd integer |

${V}_{C}$ | Capacitor voltage | $s$ | Positive odd integer with condition $z>s$ |

${V}_{in}$ | Input voltage | $b$ | Positive odd integer |

${V}_{out}$ | Output voltage | $V$ | The candidate Lyapunov’s function |

${t}_{0}$ | Initial time | $\mu $ | Positive constant |

${t}_{s}$ | Settling time | $\alpha $ | Changes in the sliding surface |

${x}_{i}\left(t\right)$ | Converter state | $\mathsf{\phi}$ | Constant coefficient |

${d}_{i}$ | External disturbances | $D$ | Unknown upper bounds of external disturbances |

${\sigma}_{i}$ | Scalar value | $\widehat{D}$ | Estimation of $D$ |

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**Figure 3.**Comparison of the behaviour of the inductor current and output voltage with the results of the method in Ref. [1].

**Figure 4.**Controller effort signals [1].

**Figure 5.**Sliding surfaces [1].

**Figure 6.**Adaptation gains [1].

**Figure 7.**Comparison of the behaviour of the inductor current and output voltage with the results of the method in Ref. [1] in case of resistance changes.

**Figure 8.**Controller effort signals [1].

**Figure 9.**Sliding surfaces [1].

**Figure 10.**Adaptation gains [1].

**Figure 11.**Comparison of the behaviour of the inductor current and output voltage with the results of the method in Ref. [1] in case of input voltage changes.

**Figure 12.**Controller effort signals [1].

**Figure 13.**Sliding surfaces [1].

**Figure 14.**Adaptation gains [1].

**Figure 15.**Comparison of the behaviour of the inductor current and output voltage with the results of the method in Ref. [1] in case of output voltage changes.

**Figure 16.**Controller effort signals [1].

**Figure 17.**Sliding surfaces [1].

**Figure 18.**Adaptation gains [1].

**Figure 20.**The behaviour of the inductor current and output voltage in the presence of measurement noise.

**Figure 24.**The behaviour of the inductor current and output voltage in the presence of measurement noise.

**Figure 28.**The behaviour of the inductor current and output voltage in case of input voltage changes and in the presence of measurement noise.

**Figure 32.**The behaviour of the inductor current and output voltage in case of output voltage changes and in the presence of measurement noise.

Scenario 1 | ||||||
---|---|---|---|---|---|---|

Proposed Approach | The Method in Ref. [1] | |||||

${x}_{1\left(t\right)}$ | ${x}_{2\left(t\right)}$ | $u\left(t\right)$ | ${x}_{1\left(t\right)}$ | ${x}_{2\left(t\right)}$ | $u\left(t\right)$ | |

Converge time | 0.022 | 0.0194 | 0.0195 | 0.113 | 0.0817 | 0.153 |

Overshoot | 0.031 | 0 | 0 | 0.33 | 1.87 | 0.01 |

Undershoot | 0 | 0 | 0 | 0 | 0 | 0 |

Chattering | NO | NO | NO | NO | NO | YES |

Scenario 2 | ||||||
---|---|---|---|---|---|---|

Proposed Approach | The Method in Ref. [1] | |||||

${x}_{1\left(t\right)}$ | ${x}_{2\left(t\right)}$ | $u\left(t\right)$ | ${x}_{1\left(t\right)}$ | ${x}_{2\left(t\right)}$ | $u\left(t\right)$ | |

Converge time | 0.0137 | 0.0195 | 0.0116 | 0.05299 | 0.0672 | 0.02213 |

Overshoot | 0 | 0 | 0 | 0.157 | 1.47 | 0 |

Undershoot | 0 | 1 | 0 | 0 | 1 | 0 |

Chattering | NO | NO | NO | NO | NO | YES |

Scenario 3 | ||||||
---|---|---|---|---|---|---|

Proposed Approach | The method in Ref. [1] | |||||

${x}_{1\left(t\right)}$ | ${x}_{2\left(t\right)}$ | $u\left(t\right)$ | ${x}_{1\left(t\right)}$ | ${x}_{2\left(t\right)}$ | $u\left(t\right)$ | |

Converge time | 0.02213 | 0.1787 | 0.0087 | 0.1302 | 0.09635 | 0.333 |

Overshoot | 0.042 | 1.28 | 0 | 0.522 | 2.41 | 0 |

Undershoot | 0 | 0 | 0 | 0 | 0 | 0 |

Chattering | NO | NO | NO | NO | NO | YES |

Scenario 4 | ||||||
---|---|---|---|---|---|---|

Proposed Approach | The method in Ref. [1] | |||||

${x}_{1\left(t\right)}$ | ${x}_{2\left(t\right)}$ | $u\left(t\right)$ | ${x}_{1\left(t\right)}$ | ${x}_{2\left(t\right)}$ | $u\left(t\right)$ | |

Converge time | 0.02774 | 0.02511 | 0.0056 | 0.1321 | 0.113 | 0.02956 |

Overshoot | 0.026 | 0.24 | 0 | 0. 511 | 2.41 | 0 |

Undershoot | 0 | 0 | 0 | 0 | 0 | 0 |

Chattering | NO | NO | NO | NO | NO | YES |

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## Share and Cite

**MDPI and ACS Style**

Alnuman, H.; Hsia, K.-H.; Sepestanaki, M.A.; Ahmed, E.M.; Mobayen, S.; Armghan, A.
Design of Continuous Finite-Time Controller Based on Adaptive Tuning Approach for Disturbed Boost Converters. *Mathematics* **2023**, *11*, 1757.
https://doi.org/10.3390/math11071757

**AMA Style**

Alnuman H, Hsia K-H, Sepestanaki MA, Ahmed EM, Mobayen S, Armghan A.
Design of Continuous Finite-Time Controller Based on Adaptive Tuning Approach for Disturbed Boost Converters. *Mathematics*. 2023; 11(7):1757.
https://doi.org/10.3390/math11071757

**Chicago/Turabian Style**

Alnuman, Hammad, Kuo-Hsien Hsia, Mohammadreza Askari Sepestanaki, Emad M. Ahmed, Saleh Mobayen, and Ammar Armghan.
2023. "Design of Continuous Finite-Time Controller Based on Adaptive Tuning Approach for Disturbed Boost Converters" *Mathematics* 11, no. 7: 1757.
https://doi.org/10.3390/math11071757