Advancing Convergence Speed of Distributed Consensus Time Synchronization Algorithms in Unmanned Aerial Vehicle Ad Hoc Networks
Abstract
:1. Introduction
1.1. Related Work
1.2. Motivations
1.3. Statement of Contributions
- A new upper bound for the constant weight in the iteration matrix of undirected graph consensus algorithms is proposed.
- The lack of a comprehensive and accurate theoretical index to describe the convergence speeds of consensus algorithms under different acceleration schemes and scenarios is highlighted.
- By establishing convex models similar to the original problem, three acceleration schemes for consensus algorithms are obtained and rigorously derived and validated through simulations. The three proposed schemes achieve faster convergence while maintaining synchronization accuracy compared with existing schemes for enhancing convergence speed in scenarios with static or known communication topologies.
- A scheme is put forward to update communication weights in real time for UANETs. This scheme minimizes the Frobenius norm of the iteration matrix. This scheme achieves the fastest convergence speed while maintaining high-precision time synchronization compared with existing schemes suitable for dynamic scenarios.
2. Preliminaries and Problem Formulation
2.1. Preliminaries
2.1.1. Network Model
2.1.2. Consensus Protocol and Matrix Theory
2.2. Problem Formulation
3. Consensus Algorithm Accelerated Convergence Scheme
3.1. Constant Communication Weight
3.2. Matrix Communication Weights
4. Performance Evaluation
4.1. Global Information Acceleration Scheme Comparison
4.1.1. Noise-Free Transmission Scenario
4.1.2. Scenario with Transmission Noise
4.1.3. Comparison of Synchronization Accuracy
4.1.4. Comparison of Convergence Speed
4.1.5. Evaluation Metric Analysis
4.2. Local Information Acceleration Scheme Comparison
5. Conclusions
6. Future Directions
- (1)
- A theoretical index that correctly and uniformly measures the convergence speeds of various consensus algorithms must be created to provide guidance for subsequent research.
- (2)
- Whether convergence speed is simultaneously related to the network topology structure, the distribution of current state values of nodes, and the current number of iterations can be explored from a projection perspective.
- (3)
- The intrinsic connection between network communication topology structure and acceleration schemes for consensus algorithms should be looked into in future work. This attempt can lead to the classification of the topology structure and proposal of optimal acceleration schemes for specific structures.
- (4)
- The differences between the sum of squares of eigenvalues of asymmetric matrices and the squares of the Frobenius norm should be determined as a basis for identifying faster acceleration schemes for distributed consensus algorithms.
- (5)
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Schemes | |
---|---|
Best constant edge weight [31] | |
Optimal symmetric edge weights [31] | |
Fastest mixing chain [49] | |
Least-mean-square weights [50] | |
Effective resistance minimization weights [51] |
Schemes | Figure 1a | Figure 1b | Figure 1c | Figure 1d | Figure 2 |
---|---|---|---|---|---|
Scheme 1 | |||||
Maximum-degree weight | |||||
Best constant edge weight | |||||
Scheme 2 | |||||
Fastest mixing chain | |||||
Optimal symmetric edge weights | |||||
Least-mean-square weights | |||||
Effective resistance minimization weights | |||||
Metropolis–Hastings weights | |||||
Lazy Metropolis weights | |||||
Scheme 3 |
Schemes | Figure 1a | Figure 1b | Figure 1c | Figure 1d | Figure 2 |
---|---|---|---|---|---|
Scheme 1 | 86 | 58 | 26 | 19 | 218 |
Maximum-degree weight | 98 | 74 | 31 | 24 | 306 |
Best constant edge weight | 62 | 42 | 26 | 17 | 177 |
Scheme 2 | 96 | 56 | 26 | 17 | 90 |
Fastest mixing chain | 62 | 47 | 26 | 17 | 65 |
Optimal symmetric edge weights | 62 | 40 | 26 | 17 | 60 |
Least-mean-square weights | 75 | 48 | 26 | 17 | 70 |
Effective resistance minimization weights | 63 | 66 | 52 | 29 | 67 |
Metropolis–Hastings weights | 98 | 74 | 31 | 19 | 153 |
Lazy Metropolis weights | 206 | 158 | 74 | 50 | 315 |
Scheme 3 | 68 | 38 | 25 | 17 | 51 |
Schemes | Figure 1a | Figure 1b | Figure 1c | Figure 1d | Figure 2 |
---|---|---|---|---|---|
Scheme 1 | 1.32 | 1.24 | 1.25 | 1.11 | 2.41 |
Maximum-degree weight | 1.33 | 1.27 | 1.27 | 1.15 | 2.78 |
Best constant edge weight | 1.41 | 1.33 | 1.25 | 1.12 | 2.96 |
Scheme 2 | 1.31 | 1.22 | 1.25 | 1.10 | 1.74 |
Fastest mixing chain | 1.41 | 1.24 | 1.25 | 1.12 | 2.75 |
Optimal symmetric edge weights | 1.41 | 1.29 | 1.25 | 1.12 | 3.09 |
Least-mean-square weights | 1.34 | 1.24 | 1.25 | 1.10 | 1.80 |
Effective resistance minimization weights | 1.42 | 1.44 | 1.29 | 1.27 | 2.01 |
Metropolis–Hastings weights | 1.33 | 1.25 | 1.27 | 1.10 | 2.05 |
Lazy Metropolis weights | 1.56 | 1.52 | 1.63 | 1.50 | 3.49 |
Scheme 3 | 1.29 | 1.19 | 1.22 | 1.10 | 1.69 |
Schemes | Figure 1a | Figure 1b | Figure 1c | Figure 1d | Figure 2 |
---|---|---|---|---|---|
Scheme 1 | 0.2197 | 0.3077 | 0.5714 | 0.7059 | 0.0842 |
Maximum-degree weight | 0.1953 | 0.2500 | 0.5000 | 0.6000 | 0.0605 |
Best constant edge weight | 0.2929 | 0.4000 | 0.5714 | 0.7500 | 0.1136 |
Scheme 2 | 0.1992 | 0.3170 | 0.5714 | 0.7368 | 0.1874 |
Fastest mixing chain | 0.2929 | 0.3636 | 0.5714 | 0.7500 | 0.2905 |
Optimal symmetric edge weights | 0.2929 | 0.4226 | 0.5714 | 0.7500 | 0.3222 |
Least-mean-square weights | 0.2501 | 0.3648 | 0.5676 | 0.7373 | 0.2299 |
Effective resistance minimization weights | 0.2893 | 0.3632 | 0.6667 | 1.0000 | 0.2536 |
Metropolis–Hastings weights | 0.1953 | 0.2500 | 0.5000 | 0.7000 | 0.1157 |
Lazy Metropolis weights | 0.0976 | 0.1250 | 0.2500 | 0.3500 | 0.0578 |
Scheme 3 | 0.2713 | 0.4357 | 0.6667 | 0.7500 | 0.2955 |
Schemes | Figure 1a | Figure 1b | Figure 1c | Figure 1d | Figure 2 |
---|---|---|---|---|---|
Scheme 1 | 0.1716 | 0.2500 | 0.4000 | 0.6000 | 0.0602 |
Maximum-degree weight | 0.1716 | 0.2500 | 0.4000 | 0.6000 | 0.0602 |
Best constant edge weight | 0.1716 | 0.2500 | 0.4000 | 0.6000 | 0.0602 |
Scheme 2 | 0.1620 | 0.2679 | 0.4000 | 0.5833 | 0.1443 |
Fastest mixing chain | 0.1716 | 0.2667 | 0.4000 | 0.6000 | 0.1700 |
Optimal symmetric edge weights | 0.1716 | 0.2679 | 0.4000 | 0.6000 | 0.1920 |
Least-mean-square weights | 0.1713 | 0.2679 | 0.4000 | 0.5840 | 0.1607 |
Effective resistance minimization weights | 0.1691 | 0.2093 | 0.4000 | 0.5714 | 0.1480 |
Metropolis–Hastings weights | 0.1716 | 0.2500 | 0.4000 | 0.5833 | 0.1082 |
Lazy Metropolis weights | 0.1716 | 0.2500 | 0.4000 | 0.5833 | 0.1082 |
Scheme 3 | 0.2208 | 0.3796 | 0.4706 | 0.6000 | 0.2365 |
Figure 1a | Figure 1b | Figure 1c | Figure 1d | Figure 2 | |
---|---|---|---|---|---|
Algebraic connectivity of Laplacian matrix | 0.5858 | 1.0000 | 2.0000 | 3.0000 | 1.3310 |
Eigenratio of Laplacian matrix | 0.1716 | 0.2500 | 0.4000 | 0.6000 | 0.0602 |
Parameters | Value or Description |
---|---|
Number of UAVs | 30 |
Spatial Area | 160 × 130 × 2 |
Mission Details | Every three UAVs took off from a different airport and sequentially passed through four target places. After hovering for five circles above each target place, the UAVs returned to their starting point. |
Communication Range | 10 km |
Maximum Flying Speed | 340 m/s |
Maximum Nodes for Communication | 29 |
Synchronization Signal Transmission Period | 1s |
Simulation Software | STK and MATLAB |
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Wu, J.; Bai, K.; Wu, H. Advancing Convergence Speed of Distributed Consensus Time Synchronization Algorithms in Unmanned Aerial Vehicle Ad Hoc Networks. Drones 2024, 8, 285. https://doi.org/10.3390/drones8070285
Wu J, Bai K, Wu H. Advancing Convergence Speed of Distributed Consensus Time Synchronization Algorithms in Unmanned Aerial Vehicle Ad Hoc Networks. Drones. 2024; 8(7):285. https://doi.org/10.3390/drones8070285
Chicago/Turabian StyleWu, Jianfeng, Kaiyuan Bai, and Huabing Wu. 2024. "Advancing Convergence Speed of Distributed Consensus Time Synchronization Algorithms in Unmanned Aerial Vehicle Ad Hoc Networks" Drones 8, no. 7: 285. https://doi.org/10.3390/drones8070285