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On Maximal Permissiveness in Partially-Observed Discrete Event Systems: Verification and Synthesis Xiang Yin and Stphane Lafortune EECS Department, University of Michigan 13th WODES, May 30-June 1, 2016 , Xian, China 0/14 X.Yin &

  1. On Maximal Permissiveness in Partially-Observed Discrete Event Systems: Verification and Synthesis Xiang Yin and StΓ©phane Lafortune EECS Department, University of Michigan 13th WODES, May 30-June 1, 2016 , Xi’an, China 0/14 X.Yin & S.Lafortune (UMich) WODES 2016 May 2016

  2. Introduction Control Engineering Perspective 2 3 𝑑 0 4 1 5 Plant G 𝑄 𝑇(𝑑) β†’ Ξ“ βˆ— 𝑇: 𝐹 𝑝 𝑄(𝑑) Supervisor β€’ 𝐹 = 𝐹 𝑑 βˆͺ 𝐹 𝑣𝑑 = 𝐹 𝑝 βˆͺ 𝐹 𝑣𝑝 βˆ— β†’ 2 E ; Disable events in 𝐹 𝑑 based on its observations β€’ Supervisor: 𝑇: 𝐹 𝑝 β€’ Closed-loop Behavior: 𝑀(𝑇/𝐻) 1/14 X.Yin & S.Lafortune (UMich) WODES 2016 May 2016

  3. Introduction β€’ 𝐻 = (π‘Œ, 𝐹, 𝑔, 𝑦 0 ) is a deterministic FSA - π‘Œ is the finite set of states - 𝐹 is the finite set of events - 𝑔: π‘Œ Γ— 𝐹 β†’ π‘Œ is the partial transition function - 𝑦 0 is the initial state β€’ Safety specification automaton: 𝑀 𝐼 βŠ† 𝑀 (𝐻) 2/14 X.Yin & S.Lafortune (UMich) WODES 2016 May 2016

  4. Introduction β€’ 𝐻 = (π‘Œ, 𝐹, 𝑔, 𝑦 0 ) is a deterministic FSA - π‘Œ is the finite set of states - 𝐹 is the finite set of events - 𝑔: π‘Œ Γ— 𝐹 β†’ π‘Œ is the partial transition function - 𝑦 0 is the initial state β€’ Safety specification automaton: 𝑀 𝐼 βŠ† 𝑀 (𝐻) βˆ— β†’ 2 𝐹 is We say that a supervisor 𝑇: 𝐹 𝑝 - Safe, if 𝑀(𝑇/𝐻) βŠ† 𝑀(𝐼) - Maximally Permissive , if for any safe supervisor 𝑇′ , we have 𝑀(𝑇/𝐻) βŠ„ 𝑀(𝑇′/𝐻) . 2/14 X.Yin & S.Lafortune (UMich) WODES 2016 May 2016

  5. Introduction β€’ 𝐻 = (π‘Œ, 𝐹, 𝑔, 𝑦 0 ) is a deterministic FSA - π‘Œ is the finite set of states - 𝐹 is the finite set of events - 𝑔: π‘Œ Γ— 𝐹 β†’ π‘Œ is the partial transition function - 𝑦 0 is the initial state β€’ Safety specification automaton: 𝑀 𝐼 βŠ† 𝑀 (𝐻) βˆ— β†’ 2 𝐹 is We say that a supervisor 𝑇: 𝐹 𝑝 - Safe, if 𝑀(𝑇/𝐻) βŠ† 𝑀(𝐼) - Maximally Permissive , if for any safe supervisor 𝑇′ , we have 𝑀(𝑇/𝐻) βŠ„ 𝑀(𝑇′/𝐻) . 𝑀(𝐼) 𝑁𝑏𝑦 1 𝑁𝑏𝑦 2 𝑀(𝐻) 2/14 X.Yin & S.Lafortune (UMich) WODES 2016 May 2016

  6. Literature Review β€’ F. Lin, and W. M. Wonham. "On observability of discrete-event systems." Inform. Sci., 44.3 (1988): 173- 198. β€’ R. Cieslak, et al. "Supervisory control of discrete-event processes with partial observations." IEEE Transactions on Automatic Control, 33.3 (1988): 249-260. - Supremal normal and controllable solution 3/14 X.Yin & S.Lafortune (UMich) WODES 2016 May 2016

  7. Literature Review β€’ F. Lin, and W. M. Wonham. "On observability of discrete-event systems." Inform. Sci., 44.3 (1988): 173- 198. β€’ R. Cieslak, et al. "Supervisory control of discrete-event processes with partial observations." IEEE Transactions on Automatic Control, 33.3 (1988): 249-260. β€’ S. Takai, and T. Ushio. "Effective computation of an β„’ 𝑛 (𝐻) -closed, controllable, and observable sublanguage arising in supervisory control." Sys. Cont. Let. 49.3 (2003): 191-200. β€’ K. Cai, R. Zhang, and W. M. Wonham. "Relative observability of discrete-event Systems and its supremal sublanguages." IEEE Trans. Automatic Control, 60.3 (2015): 659-670. - Supremal normal and controllable solution - Solutions larger than supremal normal and controllable solution 3/14 X.Yin & S.Lafortune (UMich) WODES 2016 May 2016

  8. Literature Review β€’ F. Lin, and W. M. Wonham. "On observability of discrete-event systems." Inform. Sci., 44.3 (1988): 173- 198. β€’ R. Cieslak, et al. "Supervisory control of discrete-event processes with partial observations." IEEE Transactions on Automatic Control, 33.3 (1988): 249-260. β€’ S. Takai, and T. Ushio. "Effective computation of an β„’ 𝑛 (𝐻) -closed, controllable, and observable sublanguage arising in supervisory control." Sys. Cont. Let. 49.3 (2003): 191-200. β€’ K. Cai, R. Zhang, and W. M. Wonham. "Relative observability of discrete-event Systems and its supremal sublanguages." IEEE Trans. Automatic Control, 60.3 (2015): 659-670. - Supremal normal and controllable solution - Solutions larger than supremal normal and controllable solution - These solutions are sound but not complete 3/14 X.Yin & S.Lafortune (UMich) WODES 2016 May 2016

  9. Literature Review β€’ F. Lin, and W. M. Wonham. "On observability of discrete-event systems." Inform. Sci., 44.3 (1988): 173- 198. β€’ R. Cieslak, et al. "Supervisory control of discrete-event processes with partial observations." IEEE Transactions on Automatic Control, 33.3 (1988): 249-260. β€’ S. Takai, and T. Ushio. "Effective computation of an β„’ 𝑛 (𝐻) -closed, controllable, and observable sublanguage arising in supervisory control." Sys. Cont. Let. 49.3 (2003): 191-200. β€’ K. Cai, R. Zhang, and W. M. Wonham. "Relative observability of discrete-event Systems and its supremal sublanguages." IEEE Trans. Automatic Control, 60.3 (2015): 659-670. - Supremal normal and controllable solution - Solutions larger than supremal normal and controllable solution - These solutions are sound but not complete β€’ N. Ben Hadj-Alouane, S. Lafortune, and F. Lin. "Centralized and distributed algorithms for on-line synthesis of maximal control policies under partial observation." Discrete Event Dynamic Systems 6.4 (1996): 379-427. β€’ X. Yin and S. Lafortune. "Synthesis of Maximally Permissive Supervisors for Partially-Observed Discrete-Event Systems." IEEE Trans. Automatic Control, 61.5 (2016): 1239-1254. - Solutions are both sound and complete - A certain class of maximal policies 3/14 X.Yin & S.Lafortune (UMich) WODES 2016 May 2016

  10. Literature Review β€’ F. Lin, and W. M. Wonham. "On observability of discrete-event systems." Inform. Sci., 44.3 (1988): 173- 198. β€’ R. Cieslak, et al. "Supervisory control of discrete-event processes with partial observations." IEEE Transactions on Automatic Control, 33.3 (1988): 249-260. β€’ S. Takai, and T. Ushio. "Effective computation of an β„’ 𝑛 (𝐻) -closed, controllable, and observable sublanguage arising in supervisory control." Sys. Cont. Let. 49.3 (2003): 191-200. β€’ K. Cai, R. Zhang, and W. M. Wonham. "Relative observability of discrete-event Systems and its supremal sublanguages." IEEE Trans. Automatic Control, 60.3 (2015): 659-670. - Supremal normal and controllable solution - Solutions larger than supremal normal and controllable solution - These solutions are sound but not complete β€’ N. Ben Hadj-Alouane, S. Lafortune, and F. Lin. "Centralized and distributed algorithms for on-line synthesis of maximal control policies under partial observation." Discrete Event Dynamic Systems 6.4 (1996): 379-427. β€’ X. Yin and S. Lafortune. "Synthesis of Maximally Permissive Supervisors for Partially-Observed Discrete-Event Systems." IEEE Trans. Automatic Control, 61.5 (2016): 1239-1254. - Solutions are both sound and complete - A certain class of maximal policies 𝑀(𝐼) 𝑁𝑏𝑦 3/14 X.Yin & S.Lafortune (UMich) WODES 2016 May 2016

  11. Problem Formulation β€’ Supervisor Verification Problem. βˆ— β†’ , 2 𝐹 , verify whether or not 𝑇 𝑆 is maximal. Given a safe supervisor 𝑇 𝑆 : 𝐹 𝑝 4/14 X.Yin & S.Lafortune (UMich) WODES 2016 May 2016

  12. Problem Formulation β€’ Supervisor Verification Problem. βˆ— β†’ , 2 𝐹 , verify whether or not 𝑇 𝑆 is maximal. Given a safe supervisor 𝑇 𝑆 : 𝐹 𝑝 β€’ Supervisor Synthesis Problem. βˆ— β†’ 2 𝐹 , find a safe supervisor 𝑇 Given a non-maximal safe supervisor 𝑇 𝑆 : 𝐹 𝑝 such that 𝑀 𝑇 𝑆 /𝐻 βŠ‚ 𝑀 𝑇/𝐻 . 4/14 X.Yin & S.Lafortune (UMich) WODES 2016 May 2016

  13. Problem Formulation β€’ Supervisor Verification Problem. βˆ— β†’ , 2 𝐹 , verify whether or not 𝑇 𝑆 is maximal. Given a safe supervisor 𝑇 𝑆 : 𝐹 𝑝 β€’ Supervisor Synthesis Problem. βˆ— β†’ 2 𝐹 , find a safe supervisor 𝑇 Given a non-maximal safe supervisor 𝑇 𝑆 : 𝐹 𝑝 such that 𝑀 𝑇 𝑆 /𝐻 βŠ‚ 𝑀 𝑇/𝐻 . Motivation: Lower bound behavior 𝑀 𝑠 β€’ ↓𝑫𝑷 , the infimal controllable and observable super-language β€’ 𝑴(𝑻 𝑺 /𝑯) = 𝑴 𝒔 Achieve both the lower bound and permissiveness β€’ 4/14 X.Yin & S.Lafortune (UMich) WODES 2016 May 2016

  14. Bipartite Transition System Information State : a set of states, 𝐽 ≔ 2 π‘Œ β€’ β€’ BTS: A bipartite transition system T w.r.t. G is a 7-tuple π‘ˆ , β„Ž π‘Žπ‘ π‘ˆ , 𝐹, Ξ“, 𝑧 0 π‘ˆ , 𝑅 π‘Ž π‘ˆ , β„Ž π‘π‘Ž ) π‘ˆ = (𝑅 𝑍 where π‘ˆ βŠ† 𝐽 is the set of Y-states; - 𝑅 𝑍 π‘ˆ βŠ† 𝐽 Γ— Ξ“ is the set of Z-states so that z = (𝐽 𝑨 , Ξ“ 𝑨 ) ; - 𝑅 π‘Ž π‘ˆ : 𝑅 𝑍 π‘ˆ Γ— Ξ“ β†’ Q π‘Ž π‘ˆ represents the unobservable reach; - β„Ž π‘π‘Ž π‘ˆ : 𝑅 π‘Ž π‘ˆ Γ— E β†’ Q 𝑍 π‘ˆ represents the observation transition; - β„Ž π‘Žπ‘ 0 0 𝑏 𝑐 𝑑 1 * + 𝑏 𝑐 𝑐 𝑏 3 1 4 3 0 , * + 4 𝑑 2 𝑑 1 𝑑 1 𝑑 2 *𝑑 1 + *𝑑 2 + 5 7 6 𝑑 2 𝑑 1 3,5 , *𝑑 1 + 3,6 , *𝑑 2 + 𝑼 𝑺 𝐹 𝑑 = 𝑑 1 , 𝑑 2 , 𝐹 𝑝 = *𝑏, 𝑐+ 5/14 X.Yin & S.Lafortune (UMich) WODES 2016 May 2016

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