This paper introduces a robust controller with a compensator for the Electric Helicopter Tail Rotor system, designed to handle time-varying uncertainties and nonlinearities. Unlike traditional methods, it uses fuzzy set theory to represent uncertainties as fuzzy numbers, enabling a flexible yet precise approach. By applying fuzzy arithmetic, membership functions, and -operations, robust solutions are derived to account for system vagueness. A two-player Nash game framework is employed to optimize controller parameters, balancing system efficacy (e.g., steady-state speed and corner-tracking) and control effort (e.g., minimizing input current). Simulation experiments validate the efficacy of this fuzzy-enhanced robust controller paired with Nash equilibrium optimization, revealing substantial improvements in Electric Helicopter Tail Rotor system responsiveness, stability, and energy efficiency-paving the way for more reliable aerial maneuvering in uncertain environments.
BashiriMBinazadehTAzhdariM (2025a) A Nash differential game approach in observer-based LQR controller design. Optimal Control Applications and Methods46(4): 1388–1401. https://doi.org/10.1002/oca.3265
2.
BashiriMBinazadehTAzhdariM (2025b) Robust control in two-player nonzero-sum LQ differential games with uncertainties: a spacecraft rendezvous case study. Franklin Open12: 100330. https://doi.org/10.1016/j.fraope.2025.100330
3.
ChenYH (2011) A new approach to the control design of fuzzy dynamical systems. Journal of Dynamic Systems, Measurement, and Control133(6): 061019. https://doi.org/10.1115/1.4004579
4.
ChenYHZhangXR (2010) Adaptive robust approximate constraint-following control for mechanical systems. Journal of the Franklin Institute347(1): 69–86. https://doi.org/10.1016/j.jfranklin.2009.10.012
5.
DingZTongLZDongFF, et al. (2025) A composite adaptive robust control approach for ball screw feed system with mismatched uncertainty. Journal of Vibration and Control10775463251324244. Available at: https://doi.org/10.1177/10775463251324244
6.
DongFFJinDZhaoXM, et al. (2022) A non-cooperative game approach to the robust control design for a class of fuzzy dynamical systems. ISA Transactions125: 119–133. https://doi.org/10.1016/j.isatra.2021.06.031
7.
HakimiARBinazadehT (2010) Robust adaptive limit cycle controller design for nonlinear time-delay systems. Circuits, Systems, and Signal Processing41(1): 57–76. https://doi.org/10.1007/s00034-021-01787-6
HornRAJohnsonCR (1985) Matrix Analysis. Cambridge University Press.
10.
KhalilHK (2002) Nonlinear Systems. Prentice Hall.
11.
LiCMLuSZhaoX, et al. (2023) Kinematic and dynamic performances of artificial swarm systems: aggregation, collision avoidance and compact formation. Transportation Research Part C: Emerging Technologies157: 104390. https://doi.org/10.1016/j.trc.2023.104390
12.
LiCMMaCChenYH, et al. (2024) Stratified game-theoretic optimization of robust control design for fuzzy dynamical systems: a hybrid Nash–Stackelberg strategy. IEEE Transactions on Systems, Man, and Cybernetics: Systems54(4): 2287–2296. https://doi.org/10.1109/tsmc.2023.3344190
13.
LiCMLiQYZhangDY, et al. (2025) Dynamic model and boundary control for mechanical systems with inequality constraints: a generalized Udwadia–Kalaba approach. Mechanical Systems and Signal Processing238: 113244. https://doi.org/10.1016/j.ymssp.2025.113244
14.
LiuXLZhenSCWangFL, et al. (2024) Adaptive robust control of the PMSM servo system with servo and performance constraints. Journal of Vibration and Control31: 4210–4222. https://doi.org/10.1177/10775463241278003
15.
SunHZhaoHHuangK, et al. (2017) A fuzzy approach for optimal robust control design of an automotive electronic throttle system. IEEE Transactions on Fuzzy Systems26(2): 694–704. https://doi.org/10.1109/tfuzz.2017.2688343
16.
SunHChenYHHuangK, et al. (2019) Controlling the differential mobile robot with system uncertainty: constraint-following and the adaptive robust method. Journal of Vibration and Control25(6): 1294–1305. https://doi.org/10.1177/1077546318821166
17.
WangFLChenKZhenSC, et al. (2024) Optimal design of Udwadia–Kalaba theory-based adaptive robust control for permanent magnet linear synchronous motor. Journal of Vibration and Control30(3-4): 616–631. https://doi.org/10.1177/10775463221149086
18.
XianYJHuangKKuangHD, et al. (2023) High-order robust control design and experiment for a class of uncertain serial manipulators. Journal of Vibration and Control29(21-22): 4896–4907. https://doi.org/10.1177/10775463221125987
19.
YangSYPeiJFLiuYY, et al. (2023) Robust tracking control design with a novel leakage-type adaptive mechanism for an uncertain lower limb exoskeleton robot. Journal of Vibration and Control29(11-12): 2681–2695. https://doi.org/10.1177/10775463221084401
20.
YuRRLuSZhuWY, et al. (2024a) Multi-stage trajectory tracking control design under comprehensive constraints based on generalized Udwadia–Kalaba theory. Ocean Engineering306: 117986. https://doi.org/10.1016/j.oceaneng.2024.117986
21.
YuRRZhaoYBWangGF, et al. (2024b) An intelligent cooperative game approach for adaptive robust control of fuzzy mechanical systems. IEEE Transactions on Fuzzy Systems32(6): 3594–3607. https://doi.org/10.1109/tfuzz.2024.3376694
22.
ZadehLA (1965) Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic. Fuzzy Sets and Systems90(2): 111–127. https://doi.org/10.1016/s0165-0114(97)00077-8
ZhangDYZhangYZhangH, et al. (2024) A generalized Udwadia–Kalaba control design for uncertain belt conveyor systems with inequality constraints. ISA Transactions151: 409–422. https://doi.org/10.1016/j.isatra.2024.05.046
25.
ZhangYLiQYChangX, et al. (2025) A generalized Udwadia–Kalaba control design with speed inequality constraints. IEEE Transactions on Cybernetics55(10): 4917–4928. https://doi.org/10.1109/TCYB.2025.3581228
26.
ZhaoHZhuZCSunH (2020) Adaptive robust control and optimal design for fuzzy unmanned helicopter tail reduction. International Journal of Fuzzy Systems22(5): 1400–1415. https://doi.org/10.1007/s40815-020-00870-5
27.
ZhuYFZhaoH (2024) Robust control design for electric helicopter tail deceleration system: fuzzy view and Stackelberg game theory-based optimization. ISA Transactions145: 51–62. https://doi.org/10.1016/j.isatra.2023.12.012
28.
ZhuYFZhaoHSunH, et al. (2020) Robust control design of electric helicopter tail reduction system: fuzzy and optimal view. Journal of Vibration and Control26(9-10): 814–829. https://doi.org/10.1177/1077546319889852
29.
ZhuYFZhaoHCaoZW, et al. (2023) Fuzzy approach-based optimal robust control for permanent magnet synchronous motor with experimental validation. Asian Journal of Control25(1): 170–189. https://doi.org/10.1002/asjc.2759
30.
ZhuZCZhaoHXianYJ, et al. (2024) Stackelberg game-based control design for fuzzy underactuated mechanical systems with inequality constraints. IEEE Transactions on Systems, Man, and Cybernetics: Systems54(10): 6345–6357. https://doi.org/10.1109/tsmc.2024.3428314
31.
ZhuYFQianDMZhaoH (2025) A novel robust control strategy with adaptive law for angle tracking of a two-degree-of-freedom helicopter. Journal of Vibration and Control10775463251355299. Available at: https://doi.org/10.1177/10775463251355299
