Mean-square exponential ultimate boundedness of networked control systems with packet-loss-adaptive event triggering and scale-normalized encoding

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Abstract

This paper investigates mean-square exponential ultimate boundedness (MSEUB) for bandwidth-limited networked control systems subject to Bernoulli packet losses and finite-bit communication. A packet-loss-adaptive, event-triggered mechanism updates the triggering threshold according to the number of consecutive packet losses. It also enforces a positive minimum inter-event time. A scale-normalized encoding–decoding method is then constructed for finite-bit output transmission. Before quantization, the sampled output is normalized using a scale retained locally at the field-side node. The finite-bit index vector representing the normalized output is transmitted through the sensor-to-controller channel subject to packet losses. A stochastic closed-loop model is established by decomposing the Bernoulli packet-reception variable into its mean and zero-mean fluctuation. Tractable conditions are then derived to ensure MSEUB under Bernoulli packet losses and finite-bit reconstruction errors. Finally, numerical simulations illustrate the closed-loop and communication performance of the proposed design.

Keywords

mean-square exponential ultimate boundedness / networked control systems / event-triggered control / finite-bit encoding / packet losses

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References

[1] Y. Zhi, X. Liu, S. He, W. Lin, W. Chen, Adaptively event-triggered H_∞ control for networked autonomous aerial vehicles control systems under deception attacks , IEEE Trans. Ind. Inf., 21 (2025), 3458-3465. https://doi.org/10.1109/TII.2025.3528560

[2] Z. Wang, M. Bennis, Y. Zhou, Graph attention-based MADRL for access control and resource allocation in wireless networked control systems, IEEE Trans. Wireless Commun., 23 (2024), 16076-16090. https://doi.org/10.1109/TWC.2024.3436906

[3] S. Sun, C. Jia, F. Han, Consensus of multi-agent system with disturbance based on dynamical event-triggered control, Electron. Res. Arch., 34 (2026), 2303-2320. https://doi.org/10.3934/era.2026104

[4] X. Jiang, J. Li, B. Li, Y. Wang, T. Huang, Output tracking and regulation of networked control systems under finite bandwidth constraints and its applications, IEEE Trans. Autom. Control, 69 (2024), 6490-6497. https://doi.org/10.1109/TAC.2024.3388296

[5] X. Wang, J. Zhang, C. Chen, J. He, Y. Ma, X. Guan, Trust-AoI-aware codesign of scheduling and control for edge-enabled IIoT systems, IEEE Trans. Ind. Inf., 20 (2024), 2833-2842. https://doi.org/10.1109/TII.2023.3299040

[6] Y. Ma, J. Guo, Y. Wang, A. Chakrabarty, H. Ahn, P. Orlik, et al., Optimal dynamic transmission scheduling for wireless networked control systems, IEEE Trans. Control Syst. Technol., 30 (2022), 2360-2376. https://doi.org/10.1109/TCST.2022.3141581

[7] M. Luan, Z. Chang, S. Mumtaz, G. Min, T. Hämäläinen, End-edge collaborative control for AoI-aware short-packet industrial cyber-physical system, IEEE J. Sel. Areas Commun., 43 (2025), 3104-3117. https://doi.org/10.1109/JSAC.2025.3574617

[8] L. Li, A. E. Kalør, P. Popovski, W. Chen, Unified timing analysis for closed-loop goal-oriented wireless communication, IEEE Trans. Wireless Commun., 24 (2025), 2814-2831. https://doi.org/10.1109/TWC.2024.3525087

[9] L. Chen, Y. Xu, M. Li, B. Hu, H. Guo, Z. Liu, Privacy-preserving lightweight time-series anomaly detection for resource-limited industrial IoT edge devices, IEEE Trans. Ind. Inf., 21 (2025), 4435-4446. https://doi.org/10.1109/TII.2025.3538127

[10] X. Fang, C. Lei, W. Feng, Y. Chen, M. Xiao, N. Ge, et al., Sensing-communication-computing-control closed-loop optimization for 6G digital twin-empowered robotic systems, IEEE J. Sel. Areas Commun., 43 (2025), 3330-3346. https://doi.org/10.1109/JSAC.2025.3574601

[11] P. Peng, W. Tao, H. Wang, H. Zhao, Intelligent perception communication protocol for satellite constellations via multi-agent deep reinforcement learning with dual-attention and delay-aware coordination, Electron. Res. Arch., 34 (2026), 5546-5579. https://doi.org/10.3934/era.2026248

[12] X. Zhu, X. Zhang, J. Wei, C. Wen, H. Lin, Periodic event-triggered quantized control for nonlinear systems with packet losses and large transmission delays, IEEE Trans. Autom. Control, 71 (2026), 3926-3941. https://doi.org/10.1109/TAC.2026.3653601

[13] Y. Hu, O. M. Kwon, S. H. Lee, H_∞ control for interval type-2 fuzzy singularly perturbed systems with multi-node round-robin protocol and packet dropouts , Fuzzy Sets Syst., 500 (2025), 109202. https://doi.org/10.1016/j.fss.2024.109202

[14] J. Cheng, W. Wei, Y. Wang, M. V. Basin, D. Zhang, Random time-space sampled-data control of T-S fuzzy reaction-diffusion neural networks with time-delayed communication scheme, IEEE Trans. Syst. Man Cybern. Syst., 55 (2025), 7964-7973. https://doi.org/10.1109/TSMC.2025.3598279

[15] W. Wang, H. Zeng, J. Liang, S. Xiao, Sampled-data-based load frequency control for power systems considering time delays, J. Franklin Inst., 362 (2025), 107477. https://doi.org/10.1016/j.jfranklin.2024.107477

[16] W. Wang, J. Liang, H. Zeng, Sampled-data-based stability and stabilization of Lurie systems, Appl. Math. Comput., 501 (2025), 129455. https://doi.org/10.1016/j.amc.2025.129455

[17] H. Zeng, K. Teo, Y. He, W. Wang, Sampled-data-based dissipative control of T-S fuzzy systems, Appl. Math. Model., 65 (2019), 415-427. https://doi.org/10.1016/j.apm.2018.08.012

[18] Z. Zhang, Q. Liu, M. H. Mamduhi, S. Hirche, Average communication rate for networked event-triggered stochastic control systems, IEEE Trans. Control Netw. Syst., 12 (2025), 1894-1905. https://doi.org/10.1109/TCNS.2025.3546788

[19] H. Sun, H. Mu, J. Sun, H. Han, J. Qiao, Output feedback synthesis for networked control systems with packet dropouts and multiple probability sampling periods: The stochastic communication protocol case, IEEE Trans. Cybern., 55 (2025), 4808-4819. https://doi.org/10.1109/TCYB.2025.3594482

[20] Z. Lu, G. Guo, Control and communication scheduling co-design for networked control systems: A survey, Int. J. Syst. Sci., 54 (2023), 189-203. https://doi.org/10.1080/00207721.2022.2097332

[21] J. Zhou, D. Xu, W. Tai, C. K. Ahn, Switched event-triggered H_∞ security control for networked systems vulnerable to aperiodic DoS attacks , IEEE Trans. Netw. Sci. Eng., 10 (2023), 2109-2123. https://doi.org/10.1109/TNSE.2023.3243095

[22] M. Hao, G. Zhuang, G. Cui, J. Xia, Observer-based dynamic event-triggered finite-time switching control for networked discrete implicit DC circuit systems under aperiodic DoS attacks, IEEE Trans. Autom. Sci. Eng., 22 (2025), 22925-22937. https://doi.org/10.1109/TASE.2025.3624969

[23] Y. Yang, K. G. Vamvoudakis, H. Modares, Y. Yin, D. C. Wunsch, Safe intermittent reinforcement learning with static and dynamic event generators, IEEE Trans. Neural Netw. Learn. Syst., 31 (2020), 5441-5455. https://doi.org/10.1109/TNNLS.2020.2967871

[24] M. Shen, X. Wang, S. Zhu, Z. Wu, T. Huang, Data-driven event-triggered adaptive dynamic programming control for nonlinear systems with input saturation, IEEE Trans. Cybern., 54 (2024), 1178-1188. https://doi.org/10.1109/TCYB.2023.3337779

[25] Y. Yang, G. Liu, W. Xie, W. Zhang, Q. Li, C. K. Ahn, Asymptotic event-based tracking design for nonlinear systems under multiple unknown control directions, IEEE Trans. Syst. Man Cybern. Syst., 55 (2025), 7604-7616. https://doi.org/10.1109/TSMC.2025.3595570

[26] G. Zhuang, Y. Pang, G. Zong, J. Xia, NN-based dynamic adaptive triggered security compensation feedback control for implicit power systems subjected to FDI attacks, IEEE Internet Things J., 12 (2025), 50493-50502. https://doi.org/10.1109/JIOT.2025.3609853

[27] C. Jia, S. Sun, F. Han, X. Liu, Hybrid impulsive synchronization control of heterogeneous neural networks under dynamic and static event-triggered conditions, Electron. Res. Arch., 34 (2026), 2607-2630. https://doi.org/10.3934/era.2026121

[28] Y. Yang, L. Tang, W. Zou, J. Guo, C. K. Ahn, Dynamic event-triggered design with fixed-time performance and input dead-zone, IEEE Trans. Circuits Syst. II Express Briefs, 69 (2022), 4344-4348. https://doi.org/10.1109/TCSII.2022.3184734

[29] H. Sun, C. Peng, X. Ge, Threshold-dependent secure event-triggered control for networked systems under denial of service attacks, IEEE Trans. Autom. Sci. Eng., 22 (2025), 11139-11149. https://doi.org/10.1109/TASE.2025.3531944

[30] C. Wu, X. Zhao, N. Zhao, N. Xu, Periodic and dynamic periodic event-triggered control for nonlinear cyber-physical systems under DoS attacks via a hybrid system approach, IEEE Trans. Control Netw. Syst., 11 (2024), 221-232. https://doi.org/10.1109/TCNS.2023.3275918

[31] H. Zeng, Z. Zhu, T. Peng, W. Wang, X. Zhang, Robust tracking control design for a class of nonlinear networked control systems considering bounded package dropouts and external disturbance, IEEE Trans. Fuzzy Syst., 32 (2024), 3608-3617. https://doi.org/10.1109/TFUZZ.2024.3377799

[32] W. Wang, J. Liang, H. Zeng, X. Zhang, Novel looped functionals in designing output feedback controllers for aperiodic sampled-data control systems, IEEE Trans. Autom. Sci. Eng., 22 (2025), 16397-16402. https://doi.org/10.1109/TASE.2025.3573304

[33] N. Lin, Q. Ling, Event-triggered quantized consensus of linear multiagent systems under asynchronous denial-of-service attacks, IEEE Trans. Autom. Control, 70 (2025), 697-704. https://doi.org/10.1109/TAC.2024.3422230

[34] X. Kang, H. Ishii, W. P. M. H. Heemels, Resilient quantized control of uncertain networked systems under DoS attacks, IEEE Trans. Control Netw. Syst., 13 (2026), 773-785. https://doi.org/10.1109/TCNS.2026.3662149

[35] J. Xue, X. Ding, A. Slowik, H. Tong, Stabilization of sampled-data networked control systems under Markovian bit flips: an entropy-assisted predictive correction scheme, IEEE Internet Things J., early access, 2026 (2026). https://doi.org/10.1109/JIOT.2026.3727448

[36] N. Huo, D. Shen, D. Ho, Encoding-decoding-based quantized learning control using spherical polar coordinates, IEEE Trans. Cybern., 55 (2025), 812-825. https://doi.org/10.1109/TCYB.2024.3496794

[37] T. Liu, D. Shen, D. Ho, On general linear encoding-decoding pairs for quantized iterative learning control, IEEE Trans. Cybern., 56 (2026), 2047-2060. https://doi.org/10.1109/TCYB.2025.3633720

[38] Y. Liu, W. Yang, C. Su, Y. Luo, X. Wang, Observer-based control of networked periodic piecewise systems with encoding-decoding mechanism, IEEE Trans. Cybern., 55 (2025), 2754-2764. https://doi.org/10.1109/TCYB.2025.3543878

[39] M. Wakaiki, Stabilization of infinite-dimensional systems under quantization and packet loss, IEEE Trans. Autom. Control, 70 (2025), 6625-6640. https://doi.org/10.1109/TAC.2025.3564223

[40] Y. Zhou, A. Polyakov, G. Zheng, Robust finite-time stabilization of linear systems with limited state quantization, Automatica, 171 (2025), 111967. https://doi.org/10.1016/j.automatica.2024.111967

[41] J. Zhou, J. Dong, S. Xu, C. K. Ahn, Input-to-state stabilization for Markov jump systems with dynamic quantization and multimode injection attacks, IEEE Trans. Syst. Man Cybern. Syst., 54 (2024), 2517-2529. https://doi.org/10.1109/TSMC.2023.3344869

[42] M. Shen, C. Wang, Q. Wang, H. Yan, G. Zong, Z. Zhu, Fault-tolerant synchronization control of switched complex networks by a proportional-integral intermediate observer approach, IEEE Trans. Cybern., 55 (2025), 4689-4698. https://doi.org/10.1109/TCYB.2025.3591393

[43] H. Zeng, K. Teo, Y. He, A new looped-functional for stability analysis of sampled-data systems, Automatica, 82 (2017), 328-331. https://doi.org/10.1016/j.automatica.2017.04.051

[44] M. Shen, C. Wang, Q. Wang, Y. Sun, G. Zong, Synchronization of fractional reaction-diffusion complex networks with unknown couplings, IEEE Trans. Netw. Sci. Eng., 11 (2024), 4503-4512. https://doi.org/10.1109/TNSE.2024.3432997

[45] Y. Yang, H. Modares, K. G. Vamvoudakis, F. L. Lewis, Composite adaptive critic design, IEEE/CAA J. Autom. Sinica, 12 (2025), 2525-2540. https://doi.org/10.1109/JAS.2025.125435

[46] X. Li, H. Hong, S. Xiong, Y. Yu, Prescribed-time mean-square synchronization of stochastic coupled inertial neural networks with time delays via pinning control, J. Nonlinear Dyn. Appl., 2 (2026), 61-75. https://doi.org/10.62762/JNDA.2026.284613