Abstract
In some practical applications modeled by discrete-event systems (DES), the observations of events may be no longer deterministic due to sensor faults/failures, packet loss, and/or measurement uncertainties. In this context, it is interesting to reconsider the infinite-step opacity (∞-SO) and K-step opacity (K-SO) of a DES under abnormal conditions as mentioned. In this paper, the authors extend the notions of ∞-SO and K-SO defined in the standard setting to the framework of nondeterministic observations (i.e., the event-observation mechanism is state-dependent and nondeterministic). Obviously, the extended notions of ∞-SO and K-SO are more general than the previous standard ones. To effectively verify them, a matrix-based current state estimator in the context of this advanced framework is constructed using the Boolean semi-tensor product (BSTP) technique. Accordingly, the necessary and sufficient conditions for verifying these two extended versions of opacity are provided as well as their complexity analysis. Finally, several examples are given to illustrate the obtained theoretical results.
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References
Mazaré L, Using unification for opacity properties, Verimag Technical Report, 2004.
Bryans J W, Koutny M, and Ryan P, Modelling opacity using Petri nets, Electronic Notes in Theoretical Computer Science, 2005, 121: 101–115.
Bryans J W, Koutny M, Mazaré L, et al., Opacity generalised to transition systems, International Journal of Information Security, 2008, 7(6): 421–435.
Saboori A and Hadjicostis C N, Notions of security and opaicty in discrete event systems, Procceedings of 46th IEEE Conference on Decision and Control, 2007, 5056–5061.
Saboori A and Hadjicostis C N, Verification of K-step opacity and analysis of its complexity, IEEE Transactions on Automation Science and Engineering, 2011, 8(3): 549–559.
Saboori A, Verification of infinite-step opacity and complexity considerations, IEEE Transactions on Automatic Control, 2012, 57(5): 1265–1269.
Saboori A and Hadjicostis C N, Verification of initial-state opacity in security applications of DES, Proceedings of the 9th International Workshop on Discrete Event Systems, 2008, 328–333.
Saboori A and Hadjicostis C N, Verification of initial-state opacity in security appications of discrete event systems, Information Sciences, 2013, 246: 115–132.
Yin X and Lafortune S, A new approach for the verification of infinite-step and K-step opacity using two-way observers, Automatica, 2017, 80: 162–171.
Lan H, Tong, Y, Guo J, et al., Comments “A new approach for the verification of infinite-step and K-step opacity using two-way observers” [Automatica, 2017, 80: 162–171], Automatica, 2021, 124: 109273.
Falcone Y and Marchand H, Enforcement and validation (at runtime) of various notions of opacity, Discrete Event Dynamic Systems, 2015, 25(4): 531–570.
Ji Y, Yin X, and Lafortune S, Enforcing opacity by insertion functions under multiple energy constraints, Automatica, 2019, 108: 108476.
Ji Y, Yin X, and Lafortune S, Opacity enforcement using nondeterministic publicly-known edit functions, IEEE Transactions on Automatic Control, 2019, 64(10): 4369–4376.
Mohajerani S, Ji Y, and Lafortune S, Compositional and abstraction-based approach for synthesis of edit functions for opacity enforcement, IEEE Transations on Automatic Control, 2020, 65(8): 3349–3364.
Yin X and Li S, Synthesis of dynamic masks for infinite-step opacity, IEEE Transactions on Automatic Control, 2020, 65(4): 1429–1441.
Liu R and Lu J, Enforcement for infinite-step opacity and K-step opacity via insertion, Automatica, 2022, 140: 110212.
Tong Y, Li Z, Seatzu C, et al., Verification of state-based opacity using Petri nets, IEEE Transations on Automatic Control, 2016, 62(6): 2823–2837.
Tong Y, Lan H, and Seatzu C, Verification of K-step and infinite-step opacity of bounded labeled Petri nets, Automatica, 2022, 140: 110221.
Yang S, Hou J, Yin X, et al., Opacity of networked supervisory control systems over insecure communication channels, IEEE Transactions on Control of Network Systems, 2021, 8(2): 884–896.
Yang J, Deng W, Jiang C, et al., Opacity of networked discrete event systems, Information Sciences, 2021, 543: 328–344.
Deng W, Qiu D, and Yang J, Fuzzy infinite-step opacity measure of discrete event systems and its applications, IEEE Transactions on Fuzzy Systems, 2020, 30(3): 885–892.
Deng W, Qiu D, and Yang J, Opacity measures of fuzzy discrete event systems, IEEE Transactions on Fuzzy Systems, 2021, 29(9): 2612–2622.
Alves M, Basilio J, Cunha A, et al., Robust supervisory control against intermittent loss of observations, IFAC Proceedings Volumes, 2014, 47(2): 294–299.
Lin F, Control of networked discrete event system: Dealing with communication delays and losses, SIAM Journal on Control and Optimization, 2014, 52(2): 1276–1298.
Xu S and Kumar R, Discrete event control under nondeterministic partial observation, Proceedings of 2009 IEEE International Conference on Automation Science and Engineering, 2009, 127–132.
Ushio T and Takai S, Nonblocking supervisory control of discrete event systems modeled by Mealy automata with nondeterministic output functions, IEEE Transactions on Automatic Control, 2016, 61(3): 799–804.
Takai S and Ushio T, Verification of codiagnosability for discrete event systems modeled by Mealy automata with nondeterministic output functions, IEEE Transactions on Automatic Control, 2012, 57(3): 798–804.
Zhou L, Shu S, and Lin F, Detectability of discrete-event systems under nondeterministic observations, IEEE Transactions on Automation Sciencce Engineering, 2021, 18(3): 1315–1327.
Yin X, Supervisor synthesis for Mealy automata with output functions: A model transformation approach, IEEE Transactions on Automatic Control, 2017, 62(5): 2576–2581.
Cheng D, An Introduction to Semi-Tensor Product of Matrices and Its Applications, World Scientific, Singapore, 2012.
Han X, Yang W, Chen X, et al., Detectability verification of probabilistic Boolean networks, Information Sciences, 2021, 548: 313–327.
Jiang D and Zhang K, Observability of Boolean control networks with time-variant delays in states, Journal of Systems Science & Complexity, 2018, 31(2): 436–445.
Han X, Chen Z, and Su R, Synthesis of minimally restrictive optimal stability-enforcing supervisors for nondeterministic discrete event systems, Systems & Control Letters, 2019, 123: 33–39.
Han X, Wang J, Li Z, et al., Revisiting state estimation and weak detectability of discrete-event systems, IEEE Transactions on Automation Science and Engineering, 2023, 20(1): 662–674.
Cassandras C G and Lafortune S, Introduction to Discrete Event Systems, 3rd Ed., Springer, Switzerland, 2021.
Wu Y C and Lafortune S, Comparative analysis of related notions of opacity in centralized and coordinated architectures, Discrete Event Dynamic Systems, 2013, 23: 307–339.
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This research was supported by the National Natural Science Foundation of China under Grant Nos. 61903274, 61873342, 61973175, and the Tianjin Natural Science Foundation of China under Grant No. 18JCQNJC74000.
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Chu, Q., Zhang, J., Han, X. et al. Infinite- and K-Step Opacity Verification of Discrete-Event Systems Under Nondeterministic Observations. J Syst Sci Complex 36, 1830–1850 (2023). https://doi.org/10.1007/s11424-023-2114-z
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DOI: https://doi.org/10.1007/s11424-023-2114-z