The main objective of this paper is to suppress vibrations while reducing the communication burden for a class of piezoelectric cantilever beams. Different from the existing partial differential equation (PDE)-based vibration control methods, which typically employ periodic or static event-triggering mechanisms, a dynamic event-triggered piecewise controller is proposed, and its practical feasibility is verified through experiments on a physical platform. Initially, a second-order PDE model is developed for a cantilever beam equipped with multiple macro-fiber composite (MFC) actuators, explicitly accounting for the spatially distributed and piecewise inputs. Subsequently, a sufficient condition is derived using the Lyapunov direct method to ensure the convergence of the spatiotemporal deformation of the closed-loop system under bounded external disturbances. Then, to further improve communication efficiency, a dynamic event-triggered mechanism is incorporated into the piecewise controller, effectively reducing the triggering frequency while rigorously avoiding Zeno behavior. Furthermore, the construction procedure of the piezoelectric cantilever beam-based control experimental platform is presented in detail. Experimental results clearly demonstrate that the proposed approach effectively achieves vibration attenuation while significantly reducing communication overhead compared with existing methods.
AlaouiSEÖzerAÖOuzahraM (2022) Boundary feedback stabilization of a novel bilinear and extensible piezoelectric beam model. Zeitschrift für Angewandte Mathematik und Physik74(1): 15. https://doi.org/10.1007/s00033-022-01908-4
ChenXSunYYuM, et al. (2025) Theoretical and experimental study of active vibration control of rotating composite laminated beams using Fx-NLMS algorithm. Mechanical Systems and Signal Processing224: 111959. https://doi.org/10.1016/j.ymssp.2024.111959
4.
CuiMLiuHJiangH, et al. (2022) Active vibration optimal control of piezoelectric cantilever beam with uncertainties. Measurement and Control55(5-6): 359–369. https://doi.org/10.1177/00202940221091244
5.
DingHYangDDingS, et al. (2025) Reprogrammable flexible piezoelectric actuator arrays with a high degree of freedom for shape morphing and locomotion. Soft Robotics12(4): 436–444. https://doi.org/10.1089/soro.2024.0099
6.
EchreshaviZFarboodMShasadeghiM, et al. (2025) Data-driven adaptive event-triggered terminal sliding mode control for nonlinear systems with prescribed performance. International Journal of Robust and Nonlinear Control36(4): 1888–1902. https://doi.org/10.1002/rnc.70238
7.
FarvandiAKarami MohammadiA (2024) Nonlinear vibration analysis of hyperelastic and dielectric microbeams with a control parameter using nonlinear normal modes. Nonlinear Dynamics113(3): 2045–2059. https://doi.org/10.1007/s11071-024-10367-7
8.
FeyRHBWoutersRMTNijmeijerH (2010) Proportional and derivative control for steady-state vibration mitigation in a piecewise linear beam system. Nonlinear Dynamics60: 535–549. https://doi.org/10.1007/s11071-009-9613-x
9.
FurtatIGushchinP (2022) Sampled-data in space nonlinear control of scalar semilinear parabolic and hyperbolic systems. Journal of the Franklin Institute359(2): 1176–1193. https://doi.org/10.1016/j.jfranklin.2021.11.010
10.
GaoSLiuJ (2022) Event-triggered vibration control for a class of flexible mechanical systems with bending deformation and torsion deformation based on PDE model. Mechanical Systems and Signal Processing164: 108255. https://doi.org/10.1016/j.ymssp.2021.108255
11.
GuYShenMParkJH, et al. (2025) Dynamic event-triggered H∞ filtering for fuzzy Markov jump systems subject to mismatched quantization. IEEE Transactions on Automation Science and Engineering22: 10639–10649. https://doi.org/10.1109/tase.2025.3527974
12.
JiJLiuJSunL (2024a) Linear matrix inequality-based vibration control of three-dimensional Euler–Bernoulli beam with external disturbance. Journal of Vibration and Control31(5-6): 851–863. https://doi.org/10.1177/10775463241234472
13.
JiNZhangQLiuJ (2024b) Event-triggered adaptive vibration control for a flexible satellite with time-varying actuator faults. Neurocomputing584: 127578. https://doi.org/10.1016/j.neucom.2024.127578
14.
LiJChenNZ (2025) Vortex-induced vibration (VIV) of flexible riser conveying severe slugging and straight flow in steady and oscillatory flows. Journal of Sound and Vibration595: 118728. https://doi.org/10.1016/j.jsv.2024.118728
15.
LiuYWangJWWuZ, et al. (2023) Robust H∞ control for semilinear parabolic distributed parameter systems with external disturbances via mobile actuators and sensors. IEEE Transactions on Cybernetics53(8): 4880–4893. https://doi.org/10.1109/TCYB.2022.3150171
16.
LiuYWuJYaoX (2024) Event-triggered adaptive fault-tolerant boundary control for flexible bionic fish tail with output constraint. IEEE Robotics and Automation Letters9(7): 6384–6391. https://doi.org/10.1109/lra.2024.3405386
17.
LiuYZhangWWangJ, et al. (2025) Event-triggered feedback control for nonlinear parabolic distributed parameter systems with time-varying delays. IEEE Transactions on Automation Science and Engineering22: 5308–5321. https://doi.org/10.1109/tase.2024.3418489
18.
LuQWangPLiuC (2022) An analytical and experimental study on adaptive active vibration control of sandwich beam. International Journal of Mechanical Sciences232: 107634. https://doi.org/10.1016/j.ijmecsci.2022.107634
19.
LyuZLiCJiaT (2025) Combined vibration control of flexible cantilever beam driven by MFC actuators and rotary motor. Acta Mechanica236(1): 305–320. https://doi.org/10.1007/s00707-024-04140-5
20.
MengTZhangYFuQ, et al. (2024) Adaptive internal model control for a flexible wing with unsteady aerodynamic loads. IEEE Transactions on Cybernetics54: 546–557. https://doi.org/10.1109/TCYB.2023.3278711
21.
PalangariMFEchreshaviZMessilemMA, et al. (2026) Two-stage event-triggered model predictive power control of EV charging stations with V2G capability. IEEE Transactions on Transportation Electrification12(1): 797–810. https://doi.org/10.1109/tte.2025.3621746
22.
ParanjapeAAGuanJChungSJ, et al. (2013) PDE boundary control for flexible articulated wings on a robotic aircraft. IEEE Transactions on Robotics29: 625–640. https://doi.org/10.1109/tro.2013.2240711
23.
PhilipRSanthoshBBalaramB, et al. (2023) Vibration control in fluid conveying pipes using NES with nonlinear damping. Mechanical Systems and Signal Processing194: 110250. https://doi.org/10.1016/j.ymssp.2023.110250
24.
RahnCD (2001) Mechatronic Control of Distributed Noise and Vibration: A Lyapunov Approach. New York: Springer.
25.
RenHJGuoBZ (2025) Uniform exponential stability of semi-discrete scheme for Euler–Bernoulli beam equation with non-collocated feedback. Systems & Control Letters197: 106017. https://doi.org/10.1016/j.sysconle.2024.106017
26.
SarabiAOShooshtariA (2023) Adaptive tracking control of a new nonlinear energy harvester system based on the cantilever beam structure. Journal of Vibration and Control30: 1971–1983. https://doi.org/10.1177/10775463231173614
27.
ShenMGuYWangQG, et al. (2023) Tighter interval estimation for discrete-time linear systems with a new dynamic triggering approach. IEEE Transactions on Instrumentation and Measurement72: 1008011. https://doi.org/10.1109/tim.2023.3300430
28.
ShenMWangXZhuS, et al. (2024) Data-driven event-triggered adaptive dynamic programming control for nonlinear systems with input saturation. IEEE Transactions on Cybernetics54(2): 1178–1188. https://doi.org/10.1109/TCYB.2023.3337779
29.
ShiYJiaY (2018) Multimodal shear wave deicing using fibre piezoelectric actuator on composite for aircraft wings. IEEE/ASME Transactions on Mechatronics23(5): 2090–2098. https://doi.org/10.1109/tmech.2018.2862433
WangBCosta-CastelloRNaJ, et al. (2023) Modeling and adaptive parameter estimation for a piezoelectric cantilever beam. IEEE Transactions on Circuits and Systems I: Regular Papers70(1): 481–491. https://doi.org/10.1109/tcsi.2022.3218189
32.
WuRYuanYXiaoY, et al. (2024) Event-triggered robust fault-tolerant control of a class of Euler–Bernoulli beam equations via sliding mode control. Nonlinear Dynamics112(8): 5795–5810. https://doi.org/10.1007/s11071-024-09341-0
33.
YangHLiuJ (2016) Distributed piezoelectric vibration control for a flexible-link manipulator based on an observer in the form of partial differential equations. Journal of Sound and Vibration363: 77–96. https://doi.org/10.1016/j.jsv.2015.11.001
34.
YaoXSunHZhaoZ, et al. (2024) Event-triggered bipartite consensus tracking and vibration control of flexible timoshenko manipulators under time-varying actuator faults. IEEE/CAA Journal of Automatica Sinica11: 1190–1201. https://doi.org/10.1109/jas.2024.124266
35.
YaoXXuXHeW, et al. (2025) Asymptotical event-based input-output constrained boundary control of flexible manipulator agents under a signed digraph. Science China Information Sciences68(4): 142201. https://doi.org/10.1007/s11432-024-4293-3
36.
ZhanPLouJChenT, et al. (2024) Dynamic hysteresis compensation and iterative learning control for underwater flexible structures actuated by macro fiber composites. Ocean Engineering298: 117242. https://doi.org/10.1016/j.oceaneng.2024.117242
37.
ZhangYLZhuMLiD, et al. (2022) Stabilization of two coupled wave equations with joint anti-damping and non-collocated control. Automatica135: 109995. https://doi.org/10.1016/j.automatica.2021.109995
38.
ZhangXYXieXLiuYJ, et al. (2025) Integral barrier Lyapunov function-based adaptive event-triggered control of flexible riser systems. IEEE Transactions on Automation Science and Engineering22: 433–441. https://doi.org/10.1109/tase.2024.3351739
39.
ZhaoXLiuZZhangS, et al. (2022a) Time-varying trajectory tracking boundary control of a flexible rotation beam based on servomechanism. IEEE Transactions on Industrial Electronics69(9): 9185–9195. https://doi.org/10.1109/tie.2021.3114726
40.
ZhaoXZhangSLiuZ, et al. (2022b) Vibration control for flexible manipulators with event-triggering mechanism and actuator failures. IEEE Transactions on Cybernetics52: 7591–7601. https://doi.org/10.1109/TCYB.2020.3041727
41.
ZhaoZCaiSMaG, et al. (2023) Vibration control of an experimental flexible manipulator against input saturation. IEEE/CAA Journal of Automatica Sinica10(5): 1340–1342.
42.
ZhenYLinYTangY (2024) Nonlinear dynamic and time-delay vibration control of axially moving shape memory alloy plate immersed in fluid. Nonlinear Dynamics112(8): 6231–6253. https://doi.org/10.1007/s11071-024-09385-2
43.
ZhengYYaoXLiuY (2025) Event-triggered vibration control of an axially moving belt system with output constraint. IEEE Transactions on Systems, Man, and Cybernetics: Systems55(7): 4854–4863. https://doi.org/10.1109/tsmc.2025.3560728
44.
ZhongJChenXYuanY, et al. (2021) LMI-based hybrid temporal-spatial differential control design for a class of Euler–Bernoulli beam system. Proceedings of the Institution of Mechanical Engineers - Part C: Journal of Mechanical Engineering Science235(24): 7859–7869. https://doi.org/10.1177/09544062211003611
45.
ZhongJZhangJChenX, et al. (2024) RBF neural network disturbance observer-based backstepping boundary vibration control for Euler–Bernoulli beam model with input saturation. ISA Transactions150: 67–76. https://doi.org/10.1016/j.isatra.2024.05.018
46.
ZhouWHuXWangX, et al. (2024) Active compound shape/vibration control of piezo-actuated variable camber wing section via hybrid feedback/feedforward control scheme. Aerospace Science and Technology151: 109268. https://doi.org/10.1016/j.ast.2024.109268
