# Bipartite Consensus Problems for Directed Signed Networks with External Disturbances

^{*}

## Abstract

**:**

## 1. Introduction

- We introduced an output variable for signed networks. This leads to the consensus issues being converted into the corresponding output stability issues. We focused on the stability instead of consensus, which provide a convenient approach to deal with consensus problems of signed networks. To be specific, for structurally balanced cases, we applied the nonsingular transformation to signed networks, in which a reduced-order system was developed and its output stability reflected the bipartite consensus of signed networks.
- Using the tools of robust ${H}_{\infty}$ control, we derived the necessary and sufficient conditions to ensure the bipartite consensus (or state stability) objective of directed signed networks under structurally balanced (or unbalanced) conditions. Moreover, the desired disturbance rejection performance was also satisfied.
- When the signed network was structurally balanced, we provided the mathematical expression for the terminal states of all agents. It is worth noting that the terminal states had a relationship with the external disturbances. When considering the structurally unbalanced signed network, the external disturbance had no effect on the terminal values of agents.

## 2. Preliminaries

**Definition**

**1.**

- (R1)
- L has a zero eigenvalue ${\lambda}_{1}=0$ and $n-1$ eigenvalues ${\lambda}_{2}$, ${\lambda}_{3}$, ⋯, ${\lambda}_{n}$ with positive real parts if and only if $\mathcal{G}$ is structurally balanced.
- (R2)
- All eigenvalues ${\lambda}_{1}$, ${\lambda}_{2}$, ⋯, ${\lambda}_{n}$ have positive real parts if and only if $\mathcal{G}$ is structurally unbalanced.

## 3. Problem Description

- (1)
- $$\underset{t\to \infty}{lim}\left(|{x}_{i}\left(t\right)|-|{x}_{j}\left(t\right)|\right)=0,\phantom{\rule{3.33333pt}{0ex}}\forall i,j\in {\mathcal{I}}_{n};$$
- (2)
- $$\underset{t\to \infty}{lim}{x}_{i}\left(t\right)=0,\phantom{\rule{3.33333pt}{0ex}}\forall i\in {\mathcal{I}}_{n}.$$

**Assumption**

**1.**

- (C1)
- ${\int}_{0}^{\infty}\omega \left(t\right)dt$ is absolutely convergent;
- (C2)
- there exists a constant vector $y\in {\mathbb{R}}^{n}$ such that$${\int}_{0}^{\infty}\omega \left(t\right)dt=y.$$

**Remark**

**1.**

**Proposition**

**1.**

- (1)
- the bipartite consensus can be achieved for the system (8) if and only if $\mathcal{G}$ is structurally balanced. Moreover, the terminal value $x(\infty )$ is given by$$x(\infty )=\left({\nu}^{T}x\left(0\right)\right){D}_{n}{1}_{n}$$
- (2)
- the state stability can be reached for the system (8) if and only if $\mathcal{G}$ is structurally unbalanced.

#### 3.1. Nonsingular Transformation

#### 3.2. Structurally Balanced Case

#### 3.3. Structurally Unbalanced Case

## 4. Main Results

**Theorem**

**1.**

- (1)
- When $\mathcal{G}$ is structurally balanced, the bipartite consensus objective (5) holds with $\left|\right|{T}_{z\omega}\left(s\right){\left|\right|}_{2-2}<\gamma $ if and only if there exists a positive definite matrix $P\in {\mathbb{R}}^{(n-1)\times (n-1)}$ satisfying the following matrix inequality:$$-{F}^{T}{L}^{T}{E}^{T}P-PELF+\frac{1}{{\gamma}^{2}}PE{E}^{T}P+{I}_{n-1}<0.$$In particular, if the external disturbance $\omega \left(t\right)$ satisfies Assumption 1, then for arbitrary initial state $x\left(0\right)\in {\mathbb{R}}^{n}$, the terminal value of the system (4) is provided by$$x(\infty )={\nu}^{T}\left(x\left(0\right)+y\right){D}_{n}{1}_{n}$$
- (2)
- When $\mathcal{G}$ is structurally unbalanced, the state stability objective (6) holds with $\left|\right|{T}_{z\omega}\left(s\right){\left|\right|}_{2-2}<\gamma $ if and only if there exists a positive definite matrix $P\in {\mathbb{R}}^{n\times n}$ satisfying the following matrix inequality:$$-{L}^{T}P-PL+\frac{1}{{\gamma}^{2}}{P}^{T}P+{I}_{n}<0.$$

**Proof.**

**Remark**

**2.**

## 5. Simulation Example

**Example**

**1.**

**Example**

**2.**

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Signed digraph ${\mathcal{G}}_{1}$; (

**b**) Signed digraph ${\mathcal{G}}_{2}$, where the symbols “+” and “−” represent the positive and negative weight of edges, respectively.

**Figure 2.**(

**a**) State evolution of the system (4) under the signed digraph ${\mathcal{G}}_{1}$; (

**b**) Energy trajectories of the output $z\left(t\right)$ and the external disturbance $\omega \left(t\right)$.

**Figure 3.**(

**a**) State evolution of the system (4) under the signed digraph ${\mathcal{G}}_{2}$; (

**b**) Energy trajectories of the output $z\left(t\right)$ and the external disturbance $\omega \left(t\right)$.

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**MDPI and ACS Style**

Huo, B.; Ma, J.; Du, M.
Bipartite Consensus Problems for Directed Signed Networks with External Disturbances. *Mathematics* **2023**, *11*, 4828.
https://doi.org/10.3390/math11234828

**AMA Style**

Huo B, Ma J, Du M.
Bipartite Consensus Problems for Directed Signed Networks with External Disturbances. *Mathematics*. 2023; 11(23):4828.
https://doi.org/10.3390/math11234828

**Chicago/Turabian Style**

Huo, Baoyu, Jian Ma, and Mingjun Du.
2023. "Bipartite Consensus Problems for Directed Signed Networks with External Disturbances" *Mathematics* 11, no. 23: 4828.
https://doi.org/10.3390/math11234828