# Event-Trigger-Based Finite-Time Privacy-Preserving Formation Control for Multi-UAV System

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## Abstract

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## 1. Introduction

- (1)
- A local (implemented independently by each UAV), deterministic, time-varying output mapping function was adopted to cope with the privacy-preserving formation control issues for a continuous-time multi-UAV system. All UAVs encode the internal states prior to their public transmission, hence the true value information of each UAV’s states can be kept indecipherable for honest-but-curious UAVs or other malicious eavesdroppers. Compared with the existing privacy-preserving methods based on incorporating noises [36] and state decomposition [15], the method has a simpler control structure and a lower computation complexity;
- (2)
- The finite-time stability theory was introduced to ensure the convergence performance of privacy-preserving formation control. Then, through the theoretical derivation, this paper obtained a settling time related to the UAVs’ initial states. Meanwhile, the convergence time obtained by the final experimental results can verify the settling time obtained by the theoretical results;
- (3)
- An event-triggered-based finite-time privacy-preserving formation controller was designed by selecting proper triggering conditions. To some extent, with the help of an event-triggered mechanism, the lower bandwidth usage and lower frequency of controller updates can be implemented. Additionally, the paper provides the convergence analysis and privacy analysis of the proposed controller, and simultaneously excludes Zeno behavior. Compared with the research of [15,37], the controller designed in this paper relieves pressure on the actuator and bandwidth;

## 2. Preliminaries

#### 2.1. Graph Theory

#### 2.2. Privacy-Preserving Based on Output Mask

**Assumption A1.**

**Remark 1.**

#### 2.3. Some Useful Lemmas

**Lemma 1**

**Lemma 2**

## 3. Problem Formulation

**Remark 2.**

**Definition 1.**

**Definition 2.**

- (1)
- there exists a finite-time T such that $\underset{t\to T}{lim}\left|{x}_{i}\left(t\right)-{x}_{j}\left(t\right)\right|=0$ and ${x}_{i}\left(t\right)={x}_{j}\left(t\right)ift\ge T$ for any initial condition ${x}_{i}\left(0\right)$ and any $i,j=1,\dots ,n$ [38];
- (2)
- there exists an output mapping called a privacy mask g in a condition whereby ${g}_{i}\left(0,{x}_{i},{\xi}_{i}\right)\ne {x}_{i},\phantom{\rule{4pt}{0ex}}\forall {x}_{i}\in {\Re}^{n},\phantom{\rule{4pt}{0ex}}i=1,\dots ,n$; $g(t,x,\xi )$ assures the indecipherability of the UAVs’ initial states; the neighborhoods of any ${x}_{i}\in {\Re}^{n}$ are not preserved by ${g}_{i}\left(0,{x}_{i},{\xi}_{i}\right)$; ${g}_{i}\left(t,{x}_{i},{\xi}_{i}\right)$ is strictly increasing with respect to ${x}_{i}$ for any certain t, and ${\xi}_{i},i=1,\dots ,n$ [44].

**Remark 3.**

## 4. Control Design with Event-Triggered Strategy

**Remark 4.**

**Theorem 1.**

**Proof.**

- (i)
- If the vector x holds

- (ii)
- Else if the vector x holds

**Theorem 2.**

**Proof.**

## 5. Simulation

**Algorithm 1.**

**Algorithm 2.**

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 8.**Privacy-preserving formation control for fifty UAVs ($x\left(0\right)$ vs. $m\left(0\right)$).

**Figure 9.**The true and masked state trajectories of all UAVs in a 2D plane (

**a**) True state; (

**b**) Masked state.

**Figure 10.**The true and masked state trajectories of all UAVs in 3D space (

**a**) True state; (

**b**) Masked state.

Literature | Privacy-Preserving | Object | PA | BC | CP | Results |
---|---|---|---|---|---|---|

[9] | Superimposed noise | discrete-time | ✓ | ✓ | average consensus | |

[10] | Encryption method | discrete-time | ✓ | leader-following | ||

[11] | Differential privacy | continuous-time | ✓ | output consensus | ||

[12] | State decomposition | discrete-time | ✓ | ✓ | average consensus | |

[13] | Node augment | discrete-time | ✓ | leader-following | ||

[35] | Quantized offset | discrete-time | ✓ | ✓ | ✓ | average consensus |

This article | Ouput mapping | continuous-time | ✓ | ✓ | ✓ | formation control |

UAV | Triggering Events | Inforamtion Transmission | ||||
---|---|---|---|---|---|---|

Algorithm 1 | Algorithm 2 | This Paper | Algorithm 1 | Algorithm 2 | This Paper | |

1 | 248 | 231 | 198 | 496 | 462 | 396 |

2 | 248 | 306 | 223 | 744 | 918 | 669 |

3 | 248 | 220 | 181 | 496 | 440 | 362 |

4 | 248 | 298 | 214 | 744 | 894 | 642 |

5 | 248 | 258 | 223 | 496 | 516 | 446 |

Total | 1240 | 1313 | 1039 | 2976 | 3230 | 2515 |

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## Share and Cite

**MDPI and ACS Style**

Yue, J.; Qin, K.; Shi, M.; Jiang, B.; Li, W.; Shi, L.
Event-Trigger-Based Finite-Time Privacy-Preserving Formation Control for Multi-UAV System. *Drones* **2023**, *7*, 235.
https://doi.org/10.3390/drones7040235

**AMA Style**

Yue J, Qin K, Shi M, Jiang B, Li W, Shi L.
Event-Trigger-Based Finite-Time Privacy-Preserving Formation Control for Multi-UAV System. *Drones*. 2023; 7(4):235.
https://doi.org/10.3390/drones7040235

**Chicago/Turabian Style**

Yue, Jiangfeng, Kaiyu Qin, Mengji Shi, Bing Jiang, Weihao Li, and Lei Shi.
2023. "Event-Trigger-Based Finite-Time Privacy-Preserving Formation Control for Multi-UAV System" *Drones* 7, no. 4: 235.
https://doi.org/10.3390/drones7040235