# Adaptive Terminal Sliding-Mode Synchronization Control with Chattering Elimination for a Fractional-Order Chaotic System

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries and System Description

#### 2.1. Fractional Calculus

**Definition**

**1**

**.**The α-order fractional integral is defined as

**Definition**

**2**

**.**The Caputo fractional derivative is defined as

**Definition**

**3**

**.**The Riemann–Liouville fractional derivative is

**Lemma**

**1**

**.**Let f be a continuously differentiable function. Then, it holds that

#### 2.2. Descriptions of the FOCSs

**Assumption**

**1.**

**Assumption**

**2.**

## 3. Main Results

**Lemma**

**2.**

**Proof.**

**Remark**

**1.**

**Lemma**

**3**

**Proof.**

**Remark**

**2.**

**Theorem**

**1.**

**Proof.**

**Remark**

**3.**

**Remark**

**4.**

## 4. Simulation Results

**Example**

**1.**

**Example**

**2.**

## 5. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Synchronization trajectory of the states of Lorenz and Chen FOCSs. (

**a**) ${x}_{1}-{y}_{1}$; (

**b**) ${x}_{2}-{y}_{2}$; (

**c**) ${x}_{3}-{y}_{3}$.

**Figure 4.**Trajectory of adaptive parameters. (

**a**) ${\widehat{\gamma}}_{i}$; (

**b**) ${\widehat{\delta}}_{i}$; (

**c**) ${\zeta}_{i}$; (

**d**) ${\mu}_{i}$.

**Figure 7.**Synchronization trajectory of states of FOCSs. (

**a**) ${x}_{1}-{y}_{1}$; (

**b**) ${x}_{2}-{y}_{2}$; (

**c**) ${x}_{3}-{y}_{3}$.

**Figure 8.**Adaptive parameters. (

**a**) ${\widehat{\gamma}}_{i}$; (

**b**) ${\widehat{\delta}}_{i}$; (

**c**) ${\zeta}_{i}$; (

**d**) ${\mu}_{i}$.

Symbol | Description |
---|---|

ATSMC | Adaptive terminal sliding-mode control |

FOCS | Fractional-order chaotic system |

SMC | Sliding-mode control |

$\mathbb{R}$ | Real number space |

$\mathrm{\Gamma}(\xb7)$ | Gamma function |

${\mathbb{R}}^{n}$ | n-dimensional vectors space |

$\left|\right|\xb7{\left|\right|}_{1}$ | First norm |

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**MDPI and ACS Style**

Wang, C.
Adaptive Terminal Sliding-Mode Synchronization Control with Chattering Elimination for a Fractional-Order Chaotic System. *Fractal Fract.* **2024**, *8*, 188.
https://doi.org/10.3390/fractalfract8040188

**AMA Style**

Wang C.
Adaptive Terminal Sliding-Mode Synchronization Control with Chattering Elimination for a Fractional-Order Chaotic System. *Fractal and Fractional*. 2024; 8(4):188.
https://doi.org/10.3390/fractalfract8040188

**Chicago/Turabian Style**

Wang, Chenhui.
2024. "Adaptive Terminal Sliding-Mode Synchronization Control with Chattering Elimination for a Fractional-Order Chaotic System" *Fractal and Fractional* 8, no. 4: 188.
https://doi.org/10.3390/fractalfract8040188