# Adaptive Memoryless Sliding Mode Control of Uncertain Rössler Systems with Unknown Time Delays

^{*}

## Abstract

**:**

## 1. Introduction

## 2. System Description and Problem Formulation

## 3. Lyapunov Exponents

- LE < 0, the orbit attracts to a fixed point or stable periodic orbit.
- LE = 0, the orbit is an eventually fixed point.
- LE > 0, the orbit is unstable chaos.

_{1}can be defined as below:

_{0}, ($\Vert \delta x(0)\Vert <<1$). If LE

_{1}is a positive number, the chaotic behavior is ensured. Consider the system (1), the numerical and experimental results of its variation were collected with time step (0.01 s), and the time delays are ${\tau}_{1}=1$, ${\tau}_{2}=2$. The dynamic simulation results of system (1) are shown in Figure 1, and the system parameters are ${\alpha}_{1}=0.2$, ${\alpha}_{2}=0.5$, ${\beta}_{1}=0.2$, ${\beta}_{2}=0.2$, $\gamma =5.7$. The initial states of Equation (1) are: ${x}_{1}\left(0\right)=3$, ${x}_{2}\left(0\right)=0.5$, ${x}_{3}\left(0\right)=6$, where the largest LE of the numerical and experimental results are LE(${x}_{1}$) = 4.4843, LE(${x}_{2}$) = 5.7846, LE(${x}_{3}$) = 7.1595, respectively. This confirms the chaotic behavior’s existence.

## 4. Robust Adaptive Control for Uncertain Chaos System with Unknown Time Delays

## 5. Adaptive Memoryless Controller Design for Sliding Motion

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

## 6. Numerical Simulations

#### 6.1. Robust Control with Matched Disturbances

#### 6.2. Robust Control with Unmatched Uncertainty

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**The system dynamics of controlled Rössler systems with unknown time delays and matched uncertainty.

**Figure 5.**The system dynamics of the controlled Rössler systems with unknown time-varying delays and unmatched uncertainties.

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**MDPI and ACS Style**

Yan, J.-J.; Kuo, H.-H.
Adaptive Memoryless Sliding Mode Control of Uncertain Rössler Systems with Unknown Time Delays. *Mathematics* **2022**, *10*, 1885.
https://doi.org/10.3390/math10111885

**AMA Style**

Yan J-J, Kuo H-H.
Adaptive Memoryless Sliding Mode Control of Uncertain Rössler Systems with Unknown Time Delays. *Mathematics*. 2022; 10(11):1885.
https://doi.org/10.3390/math10111885

**Chicago/Turabian Style**

Yan, Jun-Juh, and Hang-Hong Kuo.
2022. "Adaptive Memoryless Sliding Mode Control of Uncertain Rössler Systems with Unknown Time Delays" *Mathematics* 10, no. 11: 1885.
https://doi.org/10.3390/math10111885