# The Synchronization of a Class of Time-Delayed Chaotic Systems Using Sliding Mode Control Based on a Fractional-Order Nonlinear PID Sliding Surface and Its Application in Secure Communication

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## Abstract

**:**

## 1. Introduction

- Using the NLFOPID sliding surface instead of conventional sliding surfaces.
- The existence of unknown time delays.
- The limits of uncertainty and disturbance are unknown.

## 2. Basic Definitions of the Fractional-Order Derivative

**Definition**

**1.**

**Definition**

**2.**

_{0}is the initial time and $\Gamma \left(\alpha \right)$ is the Gamma function, which is defined as follows:

**Definition**

**3.**

**Remark**

**1.**

**Definition**

**4.**

**Definition**

**5.**

## 3. The System Descriptor Equations

**Definition**

**6.**

**Assumption**

**1.**

**Assumption**

**2.**

## 4. The Sliding Mode Control Approach Based on NLFPID Controllers

## 5. Stability Analysis of the Proposed Mechanism

**Theorem**

**1.**

**Proof.**

## 6. Simulation Results

## 7. Application in Secure Communication

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Synchronization of jerk systems using the proposed mechanism and applying the control signal at t = 5 s.

Root Mean Squre Error | |
---|---|

${M}_{1}\mathrm{and}{R}_{1}$ | ${M}_{2}\mathrm{and}{R}_{2}$ |

0.041544 | 0.025857 |

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**MDPI and ACS Style**

Rasouli, M.; Zare, A.; Hallaji, M.; Alizadehsani, R.
The Synchronization of a Class of Time-Delayed Chaotic Systems Using Sliding Mode Control Based on a Fractional-Order Nonlinear PID Sliding Surface and Its Application in Secure Communication. *Axioms* **2022**, *11*, 738.
https://doi.org/10.3390/axioms11120738

**AMA Style**

Rasouli M, Zare A, Hallaji M, Alizadehsani R.
The Synchronization of a Class of Time-Delayed Chaotic Systems Using Sliding Mode Control Based on a Fractional-Order Nonlinear PID Sliding Surface and Its Application in Secure Communication. *Axioms*. 2022; 11(12):738.
https://doi.org/10.3390/axioms11120738

**Chicago/Turabian Style**

Rasouli, Mohammad, Assef Zare, Majid Hallaji, and Roohallah Alizadehsani.
2022. "The Synchronization of a Class of Time-Delayed Chaotic Systems Using Sliding Mode Control Based on a Fractional-Order Nonlinear PID Sliding Surface and Its Application in Secure Communication" *Axioms* 11, no. 12: 738.
https://doi.org/10.3390/axioms11120738