This paper investigates the zonotopic set-membership estimation (ZSME) problem for a class of discrete-time complex networks subject to weighted try-once-discard (WTOD) protocol and logarithmic quantization, where the external disturbances are unknown-but-bounded. To effectively save network resources, signals are first quantized by a quantizer before being transmitted into the network, and theWTOD protocol is introduced to schedule the transmission sequence, thereby alleviating the burden of network communication. This paper aims to develop a ZSME algorithm, which is based on the quantized and protocol-scheduled measurement signals, ensuring that the actual state of each node is enclosed in a zonotope at each moment. Furthermore, by designing a correction matrix, the size of the above zonotope is minimized. Additionally, sufficient conditions for ensuring the boundedness of the developed zonotopes are provided. Finally, the effectiveness of the proposed ZSME algorithm is verified through a simulation example.
- Open Access
- Article
Zonotopic Set-Membership Estimation for Complex Networks Under Measurement Quantization and Weighted Try-Once-Discard Protocol
- Yuqing Zhang,
- Dan Liu *
Author Information
Received: 16 May 2026 | Revised: 12 Jun 2026 | Accepted: 25 Jul 2026 | Published: 17 Aug 2026
Abstract
Keywords
zonotopic set-membership estimation | complex network | weighted try-once-discard protocol | measurement quantization
References
- 1.
Albert, R.; Barabasi, A.L. Statistical Mechanics of Complex Networks. Rev. Mod. Phys. 2002, 74, 47–97.
- 2.
Arenas, A.; Diaz-Guilera, A.; Kurths, J.; et al. Synchronization in Complex Networks. Phys. Rep. 2008, 469, 93–153.
- 3.
Boccaletti, S.; Latora, V.; Moreno, Y.; et al. Complex Networks: Structure and Dynamics. Phys. Rep. 2006, 424, 175–308.
- 4.
Chen, Y.; Wang, Z.; Shen, B.; et al. Exponential Synchronization for Delayed Dynamical Networks via Intermittent Control: Dealing with Actuator Saturations. IEEE Trans. Neural Netw. Learn. Syst. 2019, 30, 1000–1012.
- 5.
He, X.; Xue, W.; Fang, H. Consistent Distributed State Estimation with Global Observability over Sensor Network. Automatica 2018, 92, 162–172.
- 6.
Zhang, B.; Liu, D.; Wang, B. Fully Distributed Event-Triggered Containment Control for Complex Networks. Int. J. Syst. Sci. 2025, 56, 3689–3701.
- 7.
Cao, L.; Zhang, W.; Guo, H. Finite-Time Cluster Synchronisation of Complex Networks with Multi-Weights and Nonlinear Couplings via Semi-Intermittent Control. Int. J. Syst. Sci. 2025, 56, 2624–2638.
- 8.
Gao, R.; Yang, G.H.; Wasly, S. Resilient Distributed State Estimation with Multi-Hop Communication. Automatica 2024, 168, 111823.
- 9.
Hasan, A.; Kuncara, I.; Widyotriatmo, A.; et al. Secure State Estimation and Control for Autonomous Ships under Cyberattacks. Syst. Sci. Control Eng. 2025, 13, 2518964.
- 10.
Zou, L.; Song, B.; Suo, J.; et al. A Survey on Outlier-Resistant State Estimation and Its Applications. Syst. Sci. Control Eng. 2025, 13, 2474471.
- 11.
Saoudi, K.; Bdirina, K.; Guesmi, K. Robust Estimation and Control of Uncertain Affine Nonlinear Systems Using Predictive Sliding Mode Control and Sliding Mode Observer. Int. J. Syst. Sci. 2024, 55, 1480–1492.
- 12.
Shen, B.; Wang, Z.; Tan, H.; et al. Robust Fusion Filtering over Multisensor Systems with Energy Harvesting Constraints. Automatica 2021, 131, 109782.
- 13.
Tan, H.; Shen, B.; Li, Q.; et al. Recursive Filtering for Stochastic Systems with Filter-and-Forward Successive Relays. IEEE/CAA J. Autom. Sin. 2024, 11, 1202–1212.
- 14.
Ge, X.; Han, Q.L.; Wang, Z. A Threshold-Parameter-Dependent Approach to Designing Distributed Event-Triggered H∞ Consensus Filters over Sensor Networks. IEEE Trans. Cybern. 2019, 49, 1148–1159.
- 15.
Han, Q.L.; Liu, Y.; Yang, F. Optimal Communication Network-Based H∞ Quantized Control with Packet Dropouts for a Class of Discrete-Time Neural Networks with Distributed Time Delay. IEEE Trans. Neural Netw. Learn. Syst. 2016, 27, 426–434.
- 16.
Zou, L.; Wang, Z.; Han, Q.L.; et al. Moving Horizon Estimation for Networked Time-Delay Systems Under Round-Robin Protocol. IEEE Trans. Autom. Control 2019, 64, 5191–5198.
- 17.
Liu, S.; Wang, Z.; Wang, L.; et al. Recursive Set-Membership State Estimation over a FlexRay Network. IEEE Trans. Syst. Man Cybern. Syst. 2022, 52, 3591–3601.
- 18.
Ding, D.; Wang, Z.; Han, Q.L. A Set-Membership Approach to Event-Triggered Filtering for General Nonlinear Systems over Sensor Networks. IEEE Trans. Autom. Control 2020, 65, 1792–1799.
- 19.
Li, Q.; Zhi, Y.; Tan, H.; et al. Protocol-Based Zonotopic State and Fault Estimation for Communication-Constrained Industrial Cyber-Physical Systems. Inf. Sci. 2023, 634, 730–743.
- 20.
Yang, H.; Yan, H.; Li, Y.; et al. Secure Zonotopic Set-Membership State Estimation for Multirate Complex Networks Under Encryption-Decryption Mechanism. IEEE Trans. Autom. Control 2024, 69, 5109–5124.
- 21.
Tan, H.; Shen, B.; Li, Q.; et al. Zonotopic Set-Membership Estimation for Time-Varying Systems Subject to Dynamical Biases and Quantization Effects. Inf. Sci. 2024, 654, 119869.
- 22.
Niu, Y.; Jia, T.; Wang, X.; et al. Output-Feedback Control Design for NCSs Subject to Quantization and Dropout. Inf. Sci. 2009, 179, 3804–3813.
- 23.
Qi, W.; Gao, M.; Ahn, C.K.; et al. Quantized Fuzzy Finite-Time Control for Nonlinear Semi-Markov Switching Systems. IEEE Trans. Circuits Syst. II Express Briefs 2020, 67, 2622–2626.
- 24.
Fu, M.; Xie, L. The Sector Bound Approach to Quantized Feedback Control. IEEE Trans. Autom. Control 2005, 50, 1698–1711.
- 25.
Fu, M.; de Souza, C.E. State Estimation for Linear Discrete-Time Systems Using Quantized Measurements. Automatica 2009, 45, 2937–2945.
- 26.
Hu, J.; Wang, Z.; Liu, G.P.; et al. Variance-Constrained Recursive State Estimation for Time-Varying Complex Networks with Quantized Measurements and Uncertain Inner Coupling. IEEE Trans. Neural Netw. Learn. Syst. 2020, 31, 1955–1967.
- 27.
Liu, D.; Wang, Z.; Liu, Y.; et al. Recursive Filtering for Stochastic Parameter Systems with Measurement Quantizations and Packet Disorders. Appl. Math. Comput. 2021, 398, 125960.
- 28.
Li, C.; Liu, Y.; Gao, M.; et al. Fault-Tolerant Formation Consensus Control for Time-Varying Multi-Agent Systems with Stochastic Communication Protocol. Int. J. Netw. Dyn. Intell. 2024, 3, 100004.
- 29.
Liu, S.; Wang, Z.; Chen, Y.; et al. Protocol-Based Unscented Kalman Filtering in the Presence of Stochastic Uncertainties
IEEE Trans. Autom. Control 2020, 65, 1303–1309. - 30.
Shen, Y.; Wang, Z.; Shen, B.; et al. Fusion Estimation for Multi-Rate Linear Repetitive Processes Under Weighted Try-Once-Discard Protocol. Inf. Fusion 2020, 55, 281–291.
- 31.
Meng, X.; Wang, H.; Li, Y.; et al. Unscented Kalman Filtering for Nonlinear Systems with Stochastic Nonlinearities Under FlexRay Protocol. Int. J. Netw. Dyn. Intell. 2025, 4, 100010.
- 32.
Shen, Y.; Wang, Z.; Shen, B.; et al. Outlier-Resistant Recursive Filtering for Multisensor Multirate Networked Systems Under Weighted Try-Once-Discard Protocol. IEEE Trans. Cybern. 2021, 51, 4897–4908.
- 33.
Geng, H.; Wang, Z.; Ma, L.; et al. Distributed Filter Design over Sensor Networks Under Try-Once-Discard Protocol: Dealing with Sensor-Bias-Corrupted Measurement Censoring. IEEE Trans. Syst. Man Cybern. Syst. 2024, 54, 3032–3043.
- 34.
Alamo, T.; Bravo, J.M.; Camacho, E.F. Guaranteed State Estimation by Zonotopes. Automatica 2005, 41, 1035–1043.
- 35.
Combastel, C. Zonotopes and Kalman Observers: Gain Optimality under Distinct Uncertainty Paradigms and Robust Convergence. Automatica 2015, 55, 265–273.
This work is licensed under a Creative Commons Attribution 4.0 International License.
Suite 4002 Level 4, 447 Collins Street, Melbourne, Victoria 3000, Australia
General Inquiries: info@sciltp.com