Consensus control of fractional-order singular MAS via adaptive pinning under switching topologies

Abstract

Fractional-order multi-agent systems with singular dynamics and time-varying network structures have gained increasing attention due to their ability to model complex interconnected processes with memory effects and algebraic constraints. In this paper, a novel adaptive pinning control framework for fractional-order singular multi-agent systems under switching topologies is proposed to achieve leader-following consensus. The proposed method simultaneously handles memory-dependent fractional dynamics, algebraic constraints from system singularity, and changing network connectivity, thereby addressing important limitations of traditional methods. By using a distributed adaptive protocol that only needs a small percentage of agents to be pinned, the technique dramatically lowers control complexity without sacrificing performance. Numerical simulations showed the framework’s practical superiority over currently existing methods in terms of convergence rate and robustness, while analytical results based on fractional Lyapunov theory established rigorous stability conditions that account for system uncertainties. Through an integrated approach to managing entangled fractional, singular, and network dynamic characteristics, these contributions enhance the control of intricate multi-agent systems. The proposed framework explicitly handles switching topologies through average dwell-time conditions and jointly connected graphs, providing rigorous stability guarantees under dynamic network changes.

Keywords

Fractional-order / Fractional-order multi-agent systems / Consensus control / Lyapunov stability and adaptive pinning strategies / Distributed multi-agent systems

References

[1] Hu W, Wen G, Rahmani A, Yu Y. Differential evolution-based parameter estimation and synchronization of heterogeneous uncertain nonlinear delayed fractional-order multi-agent systems with unknown leader. Nonlinear Dyn. 2019; 97: 1087-1105. https://doi.org/10.1007/s11071-019-05034-1

[2] Bai J, Wen G, Rahmani A. Leaderless consensus for the fractional-order nonlinear multi-agent systems under directed interaction topology. Int J Syst Sci. 2018; 49(5): 954-963.

[3] Yaghoubi Z, Talebi HA. Cluster consensus and cluster formation for nonlinear fractional-order multi-agent systems. Int J Dyn. 2024; 12(2): 586-599.

[4] Yan Z, Yang B, Luo M, Hu M. Finite-Time Partial Component Consensus for Nonlinear Leader-Following Multi-Agent Systems. Mathematics. 2024;(2227-7390): 12(22). https://doi.org/10.3390/math12223552

[5] Yan X, Li K, Yang C, Zhuang J, Cao J. Consensus of fractional-order multi-agent systems via observer-based boundary control. IEEE Trans Network Sci Eng. https://doi.org/10.1109/TNSE.2024.3371058

[6] Pan H, Yu X, Guo L. Admissible leader-following consensus of fractional-order singular multiagent system via observer-based protocol. IEEE Transactions on Circuits and Systems II: Express Briefs. 2018; 66(8): 1406-1410. https://doi.org/10.1109/TCSII.2018.2878619

[7] Yu Z, Jiang H, Hu C, Yu J. Leader-following consensus of fractional-order multi-agent systems via adaptive pinning control. Int J Control. 2015; 88(9): 1746-1756. https://doi.org/10.1080/00207179.2015.1015807

[8] An C, Su H, Chen S. Leader-following exponential consensus for higher-order fractional-order mass with mismatched disturbances and event-triggered input. IEEE Transactions on Systems, Man, and Cybernetics: Systems. 2023; 53(10): 6448-6459. https://doi.org/10.1109/TSMC.2023.3282267

[9] Gong P, Lan W. Adaptive robust tracking control for uncertain nonlinear fractional-order multi-agent systems with directed topologies. Automatica. 2018; 92: 92-99. https://doi.org/10.1016/j.automatica.2018.02.010

[10] Pan H, Liu Z, Na C. Finite-time output leader-following consensus of fractional-order linear multi-agent systems. In 2019 Chinese Control Conference (CCC). 2019; pp. 958-963. IEEE. https://doi.org/10.23919/ChiCC.2019.8865987

[11] Bai J, Wen G, Rahmani A, Yu Y. Distributed consensus tracking for the fractional-order multi-agent systems based on the sliding mode control method. Neurocomputing. 2017; 235: 210-216. https://doi.org/10.1016/j.neucom.2016.12.066

[12] Jin Y, Xu Y, Liu G, Jin Z, Li H. Event-triggered distributed sliding mode control of fractional-order nonlinear multi-agent systems. Symmetry. 2023; 15(6): 1247. https://doi.org/10.3390/sym15061247

[13] Li B, Xu B, Zhang D. Leader-Following Group Consensus of Fractional-Order Multiagent Systems via a Dynamic Event-Triggered Control Strategy. IEEE Transactions on Control of Network Systems. 2023; 11(3): 1275-1286.

[14] Chen Y, Wen G, Peng Z, Huang T, Yu Y. Necessary and sufficient conditions for group consensus of fractional multiagent systems under fixed and switching topologies via pinning control. IEEE Trans Cybern. 2019; 51(1): 28-39. https://doi.org/10.1109/tcyb.2019.2918549

[15] Zhang X, Zheng S, Ahn CK, Xie Y. Adaptive neural consensus for fractional-order multi-agent systems with faults and delays. IEEE Trans Neural Netw. 2022; 34(10): 7873-7886. https://doi.org/10.1109/tnnls.2022.3146889

[16] Amirkhani A, Barshooi AH. Consensus in multi-agent systems: a review. Artificial Intelligence Review. 2022; 55(5): 3897-3935.

[17] Luo L, Mi W, Zhong S. Adaptive consensus control of fractional multi-agent systems by distributed event-triggered strategy. Nonlinear Dyn. 2020; 100(2): 1327-1341. https://doi.org/10.1007/s11071-020-05586-7

[18] Kandasamy J, Ramachandran R, Veerasamy V, Irudayaraj AXR. Distributed leader-follower based adaptive consensus control for networked microgrids. Appl Energy. 2024; 353: 122083. https://doi.org/10.1016/j.apenergy.2023.122083

[19] Pu X, Cheng Z. Couple-group consensus of heterogeneous fractional multi-agent systems with pinning and self-adaptive controller. In 2022 34th Chinese Control and Decision Conference (CCDC). IEEE. 2022; pp. 3357-3362. https://doi.org/10.1109/CCDC55256.2022.10034347

[20] Yang J, Luo W, Yi H, Xu W. Adaptive consensus control of nonlinear fractional-order multi-agent systems with a leader. In 2019, 3rd International Symposium on Autonomous Systems (ISAS). IEEE. 2019; pp. 528-533. https://doi.org/10.1109/ISASS.2019.8757753

[21] Almeida R, Girejko E, Hristova S, Malinowska AB. Leader-following consensus for fractional multi-agent systems. Adv Differ Equ. 2019; 1-15.

[22] Shahamatkhah, E., Tabatabaei, M. (2018). Leader-following consensus of discrete-time fractional-order multi-agent systems. Chinese Physics B, 27(1), 010701. https://doi.org/10.1088/1674-1056/27/1/010701

[23] Gong P, Wang K, Lan W. Fully distributed robust consensus control of multi-agent systems with heterogeneous unknown fractional-order dynamics. Int J Syst Sci 2019; 50(10): 1902-1919. https://doi.org/10.1080/00207721.2019.1645913

[24] Lv S, Pan M, Li X, Cai W, Lan T, Li B. Consensus control of fractional-order multi-agent systems with time delays via fractional-order iterative learning control. 2019; IEEE Access: 7, 159731-159742.

[25] Zhang Y, Wen G, Rahmani A, Peng Z, Wen S. Observer-Based Intra-Cluster Lag Consensus of Multi-Agent Systems via Intermittent Adaptive Pinning Control. J Syst Sci Complex. 2023; 36(5): 1809-1829.

[26] Li Z, Gao L, Chen W, Xu Y. Distributed adaptive cooperative tracking of uncertain nonlinear fractional-order multi-agent systems. IEEE/CAA J Autom Sin. 2019; 7(1): 292-300. https://doi.org/10.1109/JAS.2019.1911858

[27] Liu Y, Xie X, Chadli M, Sun J. Leader-less consensus control of fractional-order nonlinear multi-agent systems with measurement sensitivity and actuator attacks. IEEE Transactions on Control of Network Systems. 2024; https://doi.org/10.1109/TCNS.2024.3395721

[28] Xiao P, Gu Z. Adaptive event-triggered consensus of fractional-order nonlinear multi-agent systems. 2021; IEEE Access: 10, 213-220. https://doi.org/10.1109/ACCESS.2021.3136892

[29] Ye Y, Su H. Leader-following consensus of nonlinear fractional-order multi-agent systems over directed networks. Nonlinear Dynamics. 2019; 96: 1391-1403. https://doi.org/10.1007/s11071-019-04861-6

[30] Wu J, Yu Y, Ren G. Leader-following formation control for discrete-time fractional stochastic multi-agent systems by event-triggered strategy. Fractal fract. 2024; 8(5): 246. https://doi.org/10.3390/fractalfract8050246

[31] Zhang Y, Wen G, Peng Z, Yu Y, Rahmani A. Group multiple lags consensus of fractional-order nonlinear leader-following multi-agent systems via adaptive control. T I Meas Control. 2019; 41(5): 1313-1322. https://ui.adsabs.harvard.edu/link_gateway/2019TIMC...41.1313Z/ doi: 10.1177/0142331218777570

[32] Hu T, He Z, Zhang X, Zhong S. Leader-following consensus of fractional-order multi-agent systems based on event-triggered control. Nonlinear Dyn. 2020; 99: 2219-2232. https://doi.org/10.3934/math.2025707

[33] Xu Y, Zhang Z, Xie X, Palhares RM, Li W, Wu Y. Consensus of Fractional-order fuzzy Multi-agent systems: exploring impulsive security protocols. IEEE Transactions on Control of Network Systems. 2025; https://doi.org/10.1109/TCNS.2025.3543754

[34] Wang Z, Xue D, Pan F. Event-triggered formation adaptive control of singular fractional order multi-agent systems with switching topologies and nonlinear dynamics. Nonlinear Dynamics. 2025; 113(9), 10143-10156. https://doi.org/10.1007/s11071-024-10782-w

[35] Yan X, Li K, Yang C, Zhuang J, Cao J. Consensus of fractional-order multi-agent systems via observer-based boundary control. IEEE Transactions on Network Science and Engineering. 2024; https://doi.org/10.1109/TNSE.2024.3371058

[36] Chen L, Liu C, Lopes AM, Zhang Z, Chen Y. Non-fragile consensus strategy for variable Fractional-order Multi-agent systems based on disturbance observer. IEEE Transactions on Control of Network Systems. 2025; https://doi.org/10.1109/TCNS.2025.3527254

[37] Ma YK, Raja MM, Shukla A, Vijayakumar V, Nisar KS, Thilagavathi K. New results on approximate controllability of fractional delay integrodifferential systems of order 1¡ r¡ 2 with Sobolev-type. Alex Eng J. 2023; 81: 501-518. https://doi.org/10.1016/j.aej.2023.09.043

[38] Mohan Raja M, Vijayakumar V, Tsai CW, Veluvolu KC. Optimal control strategies and continuous dependence for stochastic hilfer fractional systems with delay: a Volterra-fredholm Integrodifferential approach. Optim Control Appl. 2025; https://doi.org/10.1002/oca.70024

[39] Alhazmi M, Aljohani AF, Taha NE, Abdel-Khalek S, Bayram M, Saber S. Application of a fractal fractional operator to nonlinear glucose-insulin systems: adomian decomposition solutions. Comput Biol Med. 2025; 196: 110453. https://doi.org/10.1016/j.compbiomed.2025.110453

[40] Alawaideh YM, Al-khamiseh BM, Ra’ed SH, Abbas S, Ghannam H, Alayed T, Popa IL. Fractional-order analysis of non-linear oscillations in a Time-dependent mass pendulum. Phys Open. 2025; 100338. https://doi.org/10.1016/j.physo.2025.100338

[41] Pang D, Zhu HJ, Jinde Cao HM, Liu S. Data-driven optimal bipartite containment control for heterogeneous multi-agent systems over coopetition networks. J Frankl Inst. 2025; 362(16): 108057. https://doi.org/10.1016/j.jfranklin.2025.108057

[42] Pang D, Wu X, Li C, Wang Z. Modeling and analysis of measles epidemic model with memory effect. Int J Control. 2025; 13: 314. https://doi.org/10.1007/s40435-025-01824-1

[43] Denghao Pang, Hao Meng, Jinde Cao, Song Liu. Group consensus protocol with input delay for HMASs in cooperative-competitive networks. Neurocomputing. 2024; 596: 127931. https://doi.org/10.1016/j.neucom.2024.127931

[44] Meng H, Pang D, Cao J, et al. Optimal bipartite consensus control for heterogeneous unknown multi-agent systems via reinforcement learning. Appl Math Comput 2024; 476: 128785. https://doi.org/10.1016/j.amc.2024.128785

[45] Pang D, Liu S, Zhao X-W. Containment control analysis of delayed nonlinear fractional-order multi-agent systems. Math Methods Appl Sci. 2024; 10354. https://doi.org/10.1002/mma.10354