# Stability Analysis of Multi-Agent Tracking Systems with Quasi-Cyclic Switching Topologies

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries and Problem Formulation

#### 2.1. Notations

#### 2.2. Graph Theory

#### 2.3. Problem Formulation

**Assumption**

**1.**

- If $r<m$, $\sigma \left({t}_{i+1}\right)\in \{{\varphi}_{r+1},{\varphi}_{m+1}\}$;
- If $r=m$, $\sigma \left({t}_{i+1}\right)\in \{{\varphi}_{1},{\varphi}_{m+1}\}$;
- If $m<r<{N}_{T}$, $\sigma \left({t}_{i+1}\right)\in \{{\varphi}_{r+1},{\varphi}_{m+1},{\varphi}_{1}\}$;
- If $r={N}_{T}$, $\sigma \left({t}_{i+1}\right)\in \{{\varphi}_{1},{\varphi}_{m+1}\}$.

## 3. Controller Design and Stability Analysis

#### 3.1. Controller Design

**Remark**

**1.**

#### 3.2. Stability Analysis

**Theorem**

**1.**

**Proof.**

## 4. A Numerical Example

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

MASs | Multi-agent Systems |

QCSS | Quasi-cyclic Switching Signals |

QCST | Quasi-cyclic Switching Topologies |

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Symbol | Description |
---|---|

${x}_{i}\left(t\right)$ | state vector of the i-th agent |

${u}_{i}\left(t\right)$ | control input of the i-th tracking agent |

$\sigma \left(t\right)$ | switching signal |

${N}_{s}$ | period length of stable subsystems |

${N}_{us}$ | period length of unstable subsystems |

${e}_{ij}\left(t\right)$ | tracking error between the i-th and the j-th tracking agent |

${e}_{i}\left(t\right)$ | tracking error between the i-th tracking agent and the target agent |

${\mathcal{N}}_{s}^{-}$ | the set of stable subsystems in the quasi-cyclic switching process |

${\mathcal{N}}_{us}^{+}$ | the set of unstable subsystems in the quasi-cyclic switching process |

Stable Subsystems | Unstable Subsystems | |||||
---|---|---|---|---|---|---|

${\varphi}_{1}$ | ${\varphi}_{2}$ | ... | ${\varphi}_{m}$ | ${\varphi}_{m+1}$ | ... | ${\varphi}_{{N}_{T}}$ |

$\varphi \left(1\right)$ | $\varphi ({d}_{1}+1)$ | ⋮ | $\varphi ({d}_{m-1}+1)$ | $\varphi ({d}_{m}+1)$ | ⋮ | $\varphi ({d}_{s-1}+1)$ |

$\varphi \left(2\right)$ | $\varphi ({d}_{1}+2)$ | $\varphi ({d}_{m-1}+2)$ | $\varphi ({d}_{m}+2)$ | $\varphi ({d}_{s-1}+2)$ | ||

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ||

$\varphi \left({d}_{1}\right)$ | $\varphi \left({d}_{2}\right)$ | $\varphi \left({d}_{m}\right)$ | $\varphi \left({d}_{m+1}\right)$ | $\varphi \left({d}_{s}\right)$ |

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**MDPI and ACS Style**

Fan, D.; Shen, H.; Dong, L.
Stability Analysis of Multi-Agent Tracking Systems with Quasi-Cyclic Switching Topologies. *Appl. Sci.* **2020**, *10*, 8889.
https://doi.org/10.3390/app10248889

**AMA Style**

Fan D, Shen H, Dong L.
Stability Analysis of Multi-Agent Tracking Systems with Quasi-Cyclic Switching Topologies. *Applied Sciences*. 2020; 10(24):8889.
https://doi.org/10.3390/app10248889

**Chicago/Turabian Style**

Fan, Dongyu, Haikuo Shen, and Lijing Dong.
2020. "Stability Analysis of Multi-Agent Tracking Systems with Quasi-Cyclic Switching Topologies" *Applied Sciences* 10, no. 24: 8889.
https://doi.org/10.3390/app10248889