# LMI-Observer-Based Stabilizer for Chaotic Systems in the Existence of a Nonlinear Function and Perturbation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Description

_{ω}is the coefficient matrix of the disturbance and B is an invertible matrix.

**Assumption**

**1.**

## 3. Main Results

#### 3.1. Systems with a Nonlinear Function and Disturbance

**Theorem**

**1.**

_{2}$\in {R}^{n\times n}$whereQ

_{1}, Q

_{2}> 0 and Q

_{1}= Q

_{1}

^{T}, Q

_{2}= Q

_{2}

^{T}such that guarantees the following LMIs:

_{1}Q

_{1}, L = Q

_{2}

^{−1}P

_{2}and M = Q

_{1}

^{−1}, the signal of control (4) guarantees that the output of the system satisfies $sup\frac{|\left|y\right|{|}_{L2}}{|\left|\omega \right|{|}_{L2}}<\gamma $ and the states of the system are asymptotically stable.

**Proof.**

#### 3.2. Systems with Nonlinear Outputs

**Theorem**

**2.**

_{1}and Q

_{2}where Q

_{1}, Q

_{2}> 0 and Q

_{1}= Q

_{1}

^{T}, Q

_{2}= Q

_{2}

^{T}, it will fulfil the following LMIs:

_{1}Q

_{1}, L = Q

_{2}

^{−1}P

_{2}and M = Q

_{1}

^{−1}, the states of the system are asymptotically stable.

**Proof.**

## 4. Simulation Results

#### 4.1. A Genesio System with a Nonlinear Function and Disturbance

#### 4.2. A Genesio System with an Output Nonlinear Function

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Karami, H.; Mobayen, S.; Lashkari, M.; Bayat, F.; Chang, A.
LMI-Observer-Based Stabilizer for Chaotic Systems in the Existence of a Nonlinear Function and Perturbation. *Mathematics* **2021**, *9*, 1128.
https://doi.org/10.3390/math9101128

**AMA Style**

Karami H, Mobayen S, Lashkari M, Bayat F, Chang A.
LMI-Observer-Based Stabilizer for Chaotic Systems in the Existence of a Nonlinear Function and Perturbation. *Mathematics*. 2021; 9(10):1128.
https://doi.org/10.3390/math9101128

**Chicago/Turabian Style**

Karami, Hamede, Saleh Mobayen, Marzieh Lashkari, Farhad Bayat, and Arthur Chang.
2021. "LMI-Observer-Based Stabilizer for Chaotic Systems in the Existence of a Nonlinear Function and Perturbation" *Mathematics* 9, no. 10: 1128.
https://doi.org/10.3390/math9101128