# Robust Consensus in a Class of Fractional-Order Multi-Agent Systems with Interval Uncertainties Using the Existence Condition of Hermitian Matrices

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. FOMAS with Interval Uncertainties

**Definition**

**1.**

**Lemma**

**1.**

## 3. Swarm Robust Asymptotic Stability (Robust Consensus)

**Definition**

**2.**

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

**Remark**

**1.**

**Lemma**

**2.**

**For**$\mathbf{0}<\mathbf{\alpha}<\mathbf{1}$ there are two matrices ${Q}_{1}={Q}_{1}{}^{*}>0$ and ${Q}_{2}={Q}_{2}{}^{*}>0$ such that

**For**$\mathbf{1}<\mathbf{\alpha}<\mathbf{2}$ there is a matrix $P={P}^{*}>0$ such that

**Theorem**

**4.**

**Case**

**1.**

**Case**

**2.**

**Proof.**

**(Sufficient):**Assume that for all ${2}^{\frac{{n}^{2}+n-2}{2}}$ matrices of ${2}^{{n}^{2}}$ vertex matrices within the vertex set defined in Equation (6) the conditions of the theorem are satisfied. Now, we will prove that the interval FOMAS is asymptoticly robust stable.

**Case**

**1.**

**Case**

**2.**

**(Necessary):**Assume that the FOMAS system is swarm robust asymptotic stable (achieves consensus) and a spanning tree is included in the graph topology G. Therefore, the system in Equation (1) achieves consensus (or is swarm robust asymptotic stable) if eigenvalues of all the matrices $A-{\lambda}_{i}F$ $\left({\lambda}_{i}\ne 0\right)$ are located in the $\mathsf{\alpha}$-Hurwitz region. Based on Lemma 2 for all ${2}^{\frac{{n}^{2}+n-2}{2}}$ matrices of ${2}^{{n}^{2}}$ vertex matrices within the vertex set defined in Equation (6), its equivalent to existence positive definite Hermitian matrices ${P}_{k}={P}_{k}{}^{*}>0\text{}\in {\u2102}^{n\times n}$ such that $\overline{\lambda}\left(\beta {P}_{k}{\widehat{A}}_{k}^{v}+{\beta}^{*}{\widehat{A}}_{k}^{{v}^{*}}{P}_{k}\right)<0$ for

**case 1**when $1<\alpha <2$ and there exist two matrices ${Q}_{k1}={Q}_{k1}{}^{*}>0\text{}\in {\u2102}^{n\times n}$ and ${Q}_{k2}={Q}_{k2}{}^{*}>0\text{}\in {\u2102}^{n\times n}$ such that $\overline{\lambda}\left(\beta ({\widehat{A}}_{k}^{v}{Q}_{k2}+{Q}_{k1}{\widehat{A}}_{k}^{{v}^{*}}\right)+{\beta}^{*}\left({\widehat{A}}_{k}^{v}{Q}_{k1}+{Q}_{k2}{\widehat{A}}_{k}^{{v}^{*}}\right))<0$ in

**case 2**when $0<\alpha <1$. □

**Theorem**

**5.**

**Case**

**1.**

**Case**

**2.**

**Proof.**

**Remark**

**2.**

## 4. Simulation Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Consensus on the orientation of three airplanes with triangle formation in Example 1 [5].

**Figure 3.**Trajectories of agents in Example 1, which are the orientations of a triangle formation of airplanes in a maneuver.

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**MDPI and ACS Style**

Riazat, M.; Azizi, A.; Naderi Soorki, M.; Koochakzadeh, A.
Robust Consensus in a Class of Fractional-Order Multi-Agent Systems with Interval Uncertainties Using the Existence Condition of Hermitian Matrices. *Axioms* **2023**, *12*, 65.
https://doi.org/10.3390/axioms12010065

**AMA Style**

Riazat M, Azizi A, Naderi Soorki M, Koochakzadeh A.
Robust Consensus in a Class of Fractional-Order Multi-Agent Systems with Interval Uncertainties Using the Existence Condition of Hermitian Matrices. *Axioms*. 2023; 12(1):65.
https://doi.org/10.3390/axioms12010065

**Chicago/Turabian Style**

Riazat, Mohammadreza, Aydin Azizi, Mojtaba Naderi Soorki, and Abbasali Koochakzadeh.
2023. "Robust Consensus in a Class of Fractional-Order Multi-Agent Systems with Interval Uncertainties Using the Existence Condition of Hermitian Matrices" *Axioms* 12, no. 1: 65.
https://doi.org/10.3390/axioms12010065