PSSE: Private Set-Valued State Estimation of Cyber-Physical Systems

Hao Liu; Yuzhe Li; Ben Niu

doi:10.1109/JAS.2025.125390

Volume 13 Issue 4

Apr. 2026

IEEE/CAA Journal of Automatica Sinica

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Article Navigation > IEEE/CAA Journal of Automatica Sinica > 2026 > 13(4): 810-821

H. Liu, Y. Li, and B. Niu, “PSSE: Private set-valued state estimation of cyber-physical systems,” IEEE/CAA J. Autom. Sinica, vol. 13, no. 4, pp. 810–821, Apr. 2026. doi: 10.1109/JAS.2025.125390

Citation:

H. Liu, Y. Li, and B. Niu, “PSSE: Private set-valued state estimation of cyber-physical systems,” IEEE/CAA J. Autom. Sinica, vol. 13, no. 4, pp. 810–821, Apr. 2026. doi: 10.1109/JAS.2025.125390

H. Liu, Y. Li, and B. Niu, “PSSE: Private set-valued state estimation of cyber-physical systems,” IEEE/CAA J. Autom. Sinica, vol. 13, no. 4, pp. 810–821, Apr. 2026. doi: 10.1109/JAS.2025.125390

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PSSE: Private Set-Valued State Estimation of Cyber-Physical Systems

doi: 10.1109/JAS.2025.125390

Hao Liu^{,
,},
Yuzhe Li^,,
Ben Niu^,

Funds: This work was partially supported by the National Natural Science Foundation of China (61703286)

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Author Bio:
Hao Liu received the B.S. degree in aircraft engineering in 2007 from Shenyang Institute of Aeronautical Engineering, M.S. degree in aircraft design in 2009 from Harbin Institute of Technology, and the Ph.D. degree in control science and engineering in 2013 from Harbin Institute of Technology, respectively. Between February 2012 and August 2012, he was a Visiting Scholar in IMT Institute for Advanced Studies Lucca, Italy. He is currently an Associate Professor at Wuhan Institute of Technology. His current research interests include attack detection, resilient control, and nonlinear estimation

Yuzhe Li is currently a Professor at the State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University. He received the B.S. degree in mechanics from Peking University in 2011 and the Ph.D. degree in electronic and computer engineering from the Hong Kong University of Science and Technology (HKUST), Hong Kong, China, in 2015. Between June 2013 and August 2013, he was a Visiting Scholar at the University of Newcastle, Australia. From September 2015 to September 2017, he was a Postdoctoral Fellow in the Department of Electrical and Computer Engineering, University of Alberta, Canada. His research interests include cyber-physical systems security, sensor power control, and networked state estimation

Ben Niu received the Ph.D. degree in control theory and control engineering from Northeastern University in 2013. He is currently a Professor at the School of Control Science and Engineering, Dalian University of Technology. His research interests include switched systems, stochastic systems, adaptive control, intelligent control, and their applications
Corresponding author: Hao Liu, e-mail: liuhao1985@wit.edu.cn
Received Date: 2024-12-08
Revised Date: 2025-01-21
Accepted Date: 2025-02-27

Abstract

Abstract

This paper investigates set-valued state estimation of cyber-physical systems (CPSs) with unknown-but-bounded (UBB) noises. Note that constrained polynomial zonotopes (CPZs) are utilized to characterize both convex and non-convex sets of noises. However, privacy issues should be taken into account since outsourcing the set-valued operations to the cloud-based node is required when collecting measurements from distributed sensors. In order to address this issue, two different set-valued estimation protocols employing partially homomorphic encryption (PHE) are proposed to guarantee the corresponding privacy. Furthermore, it is proved that the proposed protocols can ensure privacy against sensor coalition, cloud coalition, and user coalition, respectively. Finally, numerical examples are provided to show the advantages and effectiveness of our results.
- Constrained polynomial zonotopes (CPZs),
- partially homomorphic encryption (PHE),
- privacy-preserving,
- set-valued estimation

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References(39)

References

[1]	V. T. H. Le, C. Stoica, T. Alamo, E. F. Camacho, and D. Dumur, “Zonotope-based set-membership estimation for multi-output uncertain systems,” in Proc. IEEE Int. Symposium on Intelligent Control, pp. 212–217, 2013.
[2]	C. Combastel, “Zonotopes and Kalman observers: Gain optimality under distinct uncertainty paradigms and robust convergence,” Automatica, vol. 55, pp. 265–273, 2015. doi: 10.1016/j.automatica.2015.03.008
[3]	Y. Wang, V. Puig, and G. Cembrano, “Set-membership approach and Kalman observer based on zonotopes for discrete-time descriptor systems,” Automatica, vol. 93, pp. 435–443, 2018. doi: 10.1016/j.automatica.2018.03.082
[4]	T. Alamo, J. M. Bravo, and E. F. Camacho, “Guaranteed state estimation by zonotopes,” Automatica, vol. 41, pp. 1035–1043, 2005. doi: 10.1016/j.automatica.2004.12.008
[5]	M. Althoff and J. J. Rath, “Comparison of guaranteed state estimators for linear time-invariant systems,” Automatica, vol. 130, Art. no. 109662, 2021. doi: 10.1016/j.automatica.2021.109662
[6]	T. Alamo, J. M. Bravo, M. J. Redondo, and E. F. Camacho, “A set-membership state estimation algorithm based on DC programming,” Automatica, vol. 44, pp. 216–224, 2008. doi: 10.1016/j.automatica.2007.05.008
[7]	B. S. Rego, G. V. Raffo, J. K. Scott, and D. M. Raimondo, “Guaranteed methods based on constrained zonotopes for set-valued state estimation of nonlinear discrete-time systems,” Automatica, vol. 111, Art. no. 108614, 2020. doi: 10.1016/j.automatica.2019.108614
[8]	B. S. Rego, J. K. Scott, D. M. Raimondo, and G. V. Raffo, “Set-valued state estimation of nonlinear discrete-time systems with nonlinear invariants based on constrained zonotopes,” Automatica, vol. 129, Art. no. 109638, 2021. doi: 10.1016/j.automatica.2021.109638
[9]	A. Alanwar, H. Said, and M. Althoff, “Distributed secure state estimation using diffusion Kalman filters and reachability analysis,” IEEE Conf. Decision and Control, pp. 4133–4139, 2019.
[10]	A. Alanwar, J. J. Rath, H. Said, K. H. Johansson, and M. Althoff, “Distributed set-based observers using diffusion strategies,” J. Franklin Institute, vol. 360, pp. 6976–6993, 2023. doi: 10.1016/j.jfranklin.2023.03.025
[11]	F. Oliva-Palomo, A. Sanchez-Orta, H. Alazki, P. Castillo, and A.-J. Muñoz-Vázquez, “Robust global observer position-yaw control based on ellipsoid method for quadrotors,” Mechanical Systems and Signal Processing, vol. 158, Art. no. 107721, 2021. doi: 10.1016/j.ymssp.2021.107721
[12]	E. Mousavinejad, X. Ge, Q.-L. Han, T. J. Lim, and L. Vlacic, “An ellipsoidal set-membership approach to distributed joint state and sensor fault estimation of autonomous ground vehicles,” IEEE/CAA J. Autom. Sinica, vol. 8, no. 6, pp. 1107–1118, 2021. doi: 10.1109/JAS.2021.1004015
[13]	J. R. Spletzer and C. J. Taylor, “A bounded uncertainty approach to multi-robot localization,” in Proc. IEEE/RSJ Int. Conf. Intelligent Robots and Systems, vol. 2, pp. 1258–1265, 2003.
[14]	M. D. Sikirić and V. Grishukhin, “Zonotopes and parallelotopes,” Southeast Asian Bulletin of Mathematics, vol. 41, pp. 197–207, 2017.
[15]	S. Ifqir, V. Puig, D. Ichalal, N. Ait-Oufroukh, and S. Mammar, “Zonotopic set-membership estimation for switched systems based on W_i-radius minimization: Vehicle application,” in Proc. 21st IFAC World Congress, pp. 1–6, 2020.
[16]	C. Trapiello and V. Puig, “A zonotopic-based watermarking design to detect replay attacks,” IEEE/CAA J. Autom. Sinica, vol. 9, no. 11, pp. 1924–1938, 2022. doi: 10.1109/JAS.2022.105944
[17]	N. Kochdumper and M. Althoff, “Sparse polynomial zonotopes: A novel set representation for reachability analysis,” IEEE Trans. Autom. Control, vol. 66, no. 9, pp. 4043–4058, 2021. doi: 10.1109/TAC.2020.3024348
[18]	B. S. Rego, D. Locatelli, D. M. Raimondo, and G. V. Raffo, “Joint state and parameter estimation based on constrained zonotopes,” Automatica, vol. 142, Art. no. 110425, 2022. doi: 10.1016/j.automatica.2022.110425
[19]	N. Kochdumper and M. Althoff, “Constrained polynomial zonotopes,” Acta Informatica, vol. 60, no. 3, pp. 279–316, 2023.
[20]	H. Guo, J. Sun, and Z.-H. Pang, “Residual-based false data injection attacks against multi-sensor estimation systems,” IEEE/CAA J. Autom. Sinica, vol. 10, no. 5, pp. 1181–1191, 2023. doi: 10.1109/JAS.2023.123441
[21]	J. Shang, M. Chen, and T. Chen, “Optimal linear encryption against stealthy attacks on remote state estimation,” IEEE Trans. Autom. Control, vol. 66, no. 8, pp. 3592–3607, 2021. doi: 10.1109/TAC.2020.3024143
[22]	W. Duo, M. C. Zhou, and A. Abusorrah, “A survey of cyber attacks on cyber physical systems: Recent advances and challenges,” IEEE/CAA J. Autom. Sinica, vol. 9, no. 5, pp. 784–800, 2022. doi: 10.1109/JAS.2022.105548
[23]	K. Teranishi, T. Sadamoto, and K. Kogiso, “Input-output history feedback controller for encrypted control with leveled fully homomorphic encryption,” IEEE Trans. Control of Network Systems, vol. 11, no. 1, pp. 271–283, 2024.
[24]	A. B. Alexandru, K. Gatsis, Y. Shoukry, S. A. Seshia, P. Tabuada, and G. J. Pappas, “Cloud-based quadratic optimization with partially homomorphic encryption,” IEEE Trans. Autom. Control, vol. 66, no. 5, pp. 2357–2364, 2021. doi: 10.1109/TAC.2020.3005920
[25]	S. Kosieradzki, X. Zhao, H. Kawase, Y. Qiu, K. Kogiso, and J. Ueda, “Secure teleoperation control using somewhat homomorphic encryption,” IFAC-PapersOnLine, vol. 55, no. 37, pp. 593–600, 2022. doi: 10.1016/j.ifacol.2022.11.247
[26]	T. Zhu, G. Li, W. Zhou, and P. S. Yu, “Differentially private data publishing and analysis: A survey,” IEEE Trans. Knowledge and Data Engineering, vol. 29, no. 8, pp. 1619–1638, 2017. doi: 10.1109/TKDE.2017.2697856
[27]	A. Alanwar, V. Gaßmann, X. He, H. Said, H. Sandberg, K. H. Johansson, and M. Althoff, “Privacy-preserving set-based estimation using partially homomorphic encryption,” European J. Control, vol. 71, Art. no. 100786, 2023. doi: 10.1016/j.ejcon.2023.100786
[28]	M. S. Darup, A. Redder, and D. E. Quevedo, “Encrypted cooperative control based on structured feedback,” IEEE Control Systems Letters, vol. 3, no. 1, pp. 37–42, 2019. doi: 10.1109/LCSYS.2018.2851152
[29]	Y. Ni, J. Wu, L. Li, and L. Shi, “Multi-party dynamic state estimation that preserves data and model privacy,” IEEE Trans. Information Forensics and Security, vol. 16, pp. 2288–2299, 2021. doi: 10.1109/TIFS.2021.3050621
[30]	C. Ying, N. Zheng, Y. Wu, M. Xu, and W.-A Zhang, “Privacy-preserving adaptive resilient consensus for multiagent systems under cyberattacks,” IEEE Trans. Industrial Informatics, vol. 20, no. 2, pp. 1630–1640, 2024. doi: 10.1109/TII.2023.3280318
[31]	X. Su, K. Fan, and W. Shi, “Privacy-preserving distributed data fusion based on attribute protection,” IEEE Trans. Industrial Informatics, vol. 15, no. 10, pp. 5765–5777, 2019. doi: 10.1109/TII.2019.2912175
[32]	R. L. Rivest, A. Shamir, and L. Adleman, “A method for obtaining digital signatures and public-key cryptosystem,” Communications of the ACM, vol. 21, no. 2, pp. 120–126, 1978. doi: 10.1145/359340.359342
[33]	K. Kogiso and T. Fujita, “Cyber-security enhancement of networked control systems using homomorphic encryption,” in Proc. 54th Conf. Decision and Control, 2015, pp. 6836–6843.
[34]	K. Yuan, P. Sang, J. Ge, B. Zhou, and C. Jia, “A timed-release e-voting scheme based on Paillier homomorphic encryption,” IEEE Trans. Sustainable Computing, vol. 9, no. 5, pp. 740–753, 2024.
[35]	O. Goldreich, Foundations of Cryptography: Volume 1, Basic Tools, Cambridge, UK: Cambridge University Press, 2007.
[36]	J. K. Scott, D. M. Raimondo, G. R. Marseglia, and R. D. Braatz, “Constrained zonotopes: A new tool for set-based estimation and fault detection,” Automatica, vol. 69, pp. 126–136, 2016. doi: 10.1016/j.automatica.2016.02.036
[37]	X. Yang and J. K. Scott, “A comparison of zonotope order reduction techniques,” Automatica, vol. 95, pp. 378–384, 2018. doi: 10.1016/j.automatica.2018.06.006
[38]	F. Farokhi, I. Shames, and N. Batterham, “Secure and private control using semi-homomorphic encryption,” Control Engineering Practice, vol. 67, pp. 13–20, 2017. doi: 10.1016/j.conengprac.2017.07.004
[39]	Z. Pan, B. Huang, and F. Liu, “Moving horizon estimation for bounded noises based on the set-membership approach,” J. Process Control, vol. 119, pp. 25–33, 2022. doi: 10.1016/j.jprocont.2022.08.015

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PSSE: Private Set-Valued State Estimation of Cyber-Physical Systems

doi: 10.1109/JAS.2025.125390

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