# Robust Control for UAV Close Formation Using LADRC via Sine-Powered Pigeon-Inspired Optimization

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## Abstract

**:**

## 1. Introduction

## 2. Close Formation Modeling

#### 2.1. Wake Vortex Model

**Remark 1.**

#### 2.2. Trailing UAV Model

## 3. Structure of the First-Order LADRC

**Remark 2.**

## 4. Robust Control System Design

#### 4.1. Control Objective

#### 4.2. Control System Design

**Remark 3.**

#### 4.3. Stability Analysis of the Control System

**Theorem 1.**

**Proof.**

## 5. Sine-Powered Pigeon-Inspired Optimization

#### 5.1. Standard PIO Algorithm

#### 5.2. SCPIO Algorithm

**Remark 4.**

#### 5.3. Construction of the Fitness Function

#### 5.4. Optimization Procedure

- Step 1.
- Initialize the SCPIO parameters, including the number of pigeons ${N}^{pig}$, the dimension of thesearch space ${D}^{pig}$, the maximum and minimum values of the map and compass factor ${r}_{max}$ and ${r}_{min}$, the iteration numbers of two operators ${N}_{{C}_{1max}}^{pig}$ and ${N}_{{C}_{2max}}^{pig}$, and the position ${\mathit{X}}_{i}^{pig}$ and velocity ${\mathit{V}}_{i}^{pig}$ of all pigeons.
- Step 2.
- Drive the close-formation simulation system using the pigeons in Step 1 to calculate the fitness function. Compare the fitness value and find the current optimal position.
- Step 3.
- Conduct the iteration. If ${N}_{C}\le {N}_{{C}_{1max}}^{pig}$, perform the improved map and compass operator to update the pigeons. Then, drive the close-formation simulation system using the updated pigeons to calculate the fitness function. Update the optimal position by comparing the new fitness values with the current optimal one. When ${N}_{{C}_{1max}}^{pig}<{N}_{C}$, perform the landmark operator to continue the similar optimization process.
- Step 4.
- Once the iteration time reaches ${N}_{{C}_{1max}}^{pig}+{N}_{{C}_{2max}}^{pig}$, terminate the algorithm and output the optimal position ${\mathit{X}}_{gbest}^{pig}$.

Algorithm 1: SCPIO. |

## 6. Simulation Results and Analysis

#### 6.1. Analysis of the Sweet Spot

#### 6.2. Implementation of Control System Optimization

#### 6.3. Tracking-Performance Validation

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**Sectional views of dimensionless velocity field induced by the wake vortex at different longitudinal positions.

**Figure 9.**Velocity, aerodynamic angles, attitude angles, and attitude rate responses of the trailing UAV.

Algorithm | Parameter | Description | Value |
---|---|---|---|

PSO | ${N}_{{C}_{max}}$ | Maximum iterative number | 50 |

${N}_{PSO}$ | Number of particles | 100 | |

${\omega}_{PSO}$ | Inertia weight | 0.4 | |

${c}_{1}$ | Self-learning factor | 2 | |

${c}_{2}$ | Group-learning factor | 2 | |

PIO, SCPIO | ${N}_{{C}_{1max}}^{pig}$ | Iteration number of the map and compass operator | 30 |

${N}_{{C}_{2max}}^{pig}$ | Iteration number of the landmark operator | 20 | |

${N}^{pig}$ | Number of pigeons | 100 | |

R | Map and compass factor | 0.4 | |

$[{r}_{min},{r}_{max}]$ | Range of the control parameter of the sine map | [0.1, 0.9] |

**Table 2.**Optimal values of each optimization algorithm (The SCPIO algorithm and its optimized results are highlighted in bold).

Channel | Control Parameters | Algorithm | Optimal Values | Fitness Value |
---|---|---|---|---|

Longitudinal | $[{K}_{{p}_{{x}_{E}}},{\omega}_{{x}_{E}},{K}_{{p}_{V}},{\omega}_{V}]$ | SCPIO | [0.0712, 0.11, 0.26, 6.12] | 60,126 |

PIO | [0.0634, 0.16, 0.23, 6.56] | 72,151 | ||

PSO | [0.0607, 0.20, 0.21, 6.81] | 78,136 | ||

Altitude | $[{K}_{{p}_{{z}_{E}}},{\omega}_{{z}_{E}},{K}_{{p}_{\theta}},{\omega}_{\theta}]$ | SCPIO | [0.06854, 0.71, 1.05, 8.21] | 139,842 |

PIO | [0.07345, 0.64, 1.16, 7.78] | 199,774 | ||

PSO | [0.07562, 0.61, 1.24, 7.46] | 239,729 | ||

Lateral | $[{K}_{{p}_{{y}_{E}}},{\omega}_{{y}_{E}},{K}_{{p}_{\varphi}},{\omega}_{\varphi}$, ${K}_{{p}_{\psi}},{\omega}_{\psi}]$ | SCPIO | [0.0254, 0.27, 0.98, 7.04, 1.10, 7.58] | 93,251 |

PIO | [0.0207, 0.30, 0.87, 7.43, 1.21, 7.42] | 134,696 | ||

PSO | [0.0318, 0.25, 1.25, 6.78, 1.00, 7.63] | 113,974 |

Channel | Control Parameters | Algorithm | Optimal Values | Fitness Value |
---|---|---|---|---|

Longitudinal | $[{K}_{{p}_{{x}_{E}}}^{PI},{K}_{{p}_{V}}^{PI}]$ | SCPIO | [0.1024, 250.7534] | 28,146 |

Altitude | $[{K}_{{p}_{{z}_{E}}}^{PI},{K}_{{i}_{{z}_{E}}}^{PI},{K}_{{p}_{\theta}}^{PI}]$ | SCPIO | [0.0021, 0.0005, 1.1568] | 39,084 |

Lateral | $[{K}_{{p}_{{y}_{E}}}^{PI},{K}_{{p}_{\varphi}}^{PI},{K}_{{p}_{\psi}}^{PI}]$ | SCPIO | [0.08791, 1.1326, 2.9736] | 10,211 |

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**MDPI and ACS Style**

Yuan, G.; Duan, H. Robust Control for UAV Close Formation Using LADRC via Sine-Powered Pigeon-Inspired Optimization. *Drones* **2023**, *7*, 238.
https://doi.org/10.3390/drones7040238

**AMA Style**

Yuan G, Duan H. Robust Control for UAV Close Formation Using LADRC via Sine-Powered Pigeon-Inspired Optimization. *Drones*. 2023; 7(4):238.
https://doi.org/10.3390/drones7040238

**Chicago/Turabian Style**

Yuan, Guangsong, and Haibin Duan. 2023. "Robust Control for UAV Close Formation Using LADRC via Sine-Powered Pigeon-Inspired Optimization" *Drones* 7, no. 4: 238.
https://doi.org/10.3390/drones7040238