A General Approach for Synthesis of Supervisors for - PowerPoint PPT Presentation

A General Approach for Synthesis of Supervisors for Partially-Observed Discrete-Event Systems Xiang Yin and Stéphane Lafortune EECS Department, University of Michigan 19th IFAC WC, August 24-29, 2014, Cape Town, South Africa 1/18 X.Yin & S.Lafortune (UMich) IFAC World Congress 2014 August 25, 2014

Introduction • Supervisory control under partial observation 2 3 0 4 𝑡 1 5 Plant G 𝑄 𝑇(𝑡) ∗ → Γ 𝑄(𝑡) 𝑇: 𝐹 𝑝 Supervisor 𝐹 = 𝐹 𝑑 ∪ 𝐹 𝑣𝑑 = 𝐹 𝑝 ∪ 𝐹 𝑣𝑝 • → Γ , where Γ ≔ {𝛿 ∈ 2 𝐹 : 𝐹 𝑣𝑑 ⊆ 𝛿} Supervisor 𝑇: 𝑄 ℒ 𝐻 • 2/18 X.Yin & S.Lafortune (UMich) IFAC World Congress 2014 August 25, 2014

System Model 𝐻 = (𝑌, 𝐹, 𝑔, 𝑦 0 ) is a deterministic FSA 𝑌 is the finite set of states; • 𝐹 is the finite set of events; • 𝑔: 𝑌 × 𝐹 → 𝑌 is the partial transition function; • 𝑦 0 is the initial state. • 3/18 X.Yin & S.Lafortune (UMich) IFAC World Congress 2014 August 25, 2014

System Model 𝐻 = (𝑌, 𝐹, 𝑔, 𝑦 0 ) is a deterministic FSA 𝑌 is the finite set of states; • 𝐹 is the finite set of events; • 𝑔: 𝑌 × 𝐹 → 𝑌 is the partial transition function; • 𝑦 0 is the initial state. • Specification automaton 𝐼 : 𝐿 = ℒ 𝐼 ⊆ ℒ (𝐻) • • Assumption : illegality is captured by states (w.l.o.g.) 𝑌 𝐼 ⊆ 𝑌 is the set of legal states 3/18 X.Yin & S.Lafortune (UMich) IFAC World Congress 2014 August 25, 2014

Problem Formulation • Existence Condition: (Controllability and Observability Theorem) There exists a supervisor such that ℒ (𝑇/𝐻) = 𝐿 if and only if 𝐿 is controllable and observable. 4/18 X.Yin & S.Lafortune (UMich) IFAC World Congress 2014 August 25, 2014

Problem Formulation • Existence Condition: (Controllability and Observability Theorem) There exists a supervisor such that ℒ (𝑇/𝐻) = 𝐿 if and only if 𝐿 is controllable and observable. • Synthesis Problem: (BSCOP 𝑛𝑏𝑦 ) ∗ → Γ such that Given a plant 𝐻 and specification 𝐼 . Find a supervisor 𝑇: 𝐹 𝑝 1). ℒ (𝑇/𝐻) ⊆ ℒ 𝐼 ; (Safety) 2). ℒ(𝑇/𝐻) ⊄ ℒ(𝑇 ′ /𝐻) , ∀ safe 𝑇 ′ . (Maximal Permissiveness) 4/18 X.Yin & S.Lafortune (UMich) IFAC World Congress 2014 August 25, 2014

Literature Survey • F. Lin, and W. M. Wonham. "On observability of discrete-event systems." Information sciences 44.3 (1988): 173-198. • R. Cieslak, et al. "Supervisory control of discrete-event processes with partial observations." IEEE Transactions on Automatic Control, 33.3 (1988): 249-260. (Initial works; Supremal normal solution) 5/18 X.Yin & S.Lafortune (UMich) IFAC World Congress 2014 August 25, 2014

Literature Survey • F. Lin, and W. M. Wonham. "On observability of discrete-event systems." Information sciences 44.3 (1988): 173-198. • R. Cieslak, et al. "Supervisory control of discrete-event processes with partial observations." IEEE Transactions on Automatic Control, 33.3 (1988): 249-260. (Initial works; Supremal normal solution) S. Takai, and T. Ushio. "Effective computation of an ℒ 𝑛 (𝐻) -closed, controllable, • and observable sublanguage arising in supervisory control." Systems & Control Letters 49.3 (2003): 191-200. • K. Cai, R. Zhang, and W. M. Wonham. "Relative observability of discrete-event Systems and its supremal sublanguages." IEEE Transactions on Automatic Control, (2014). (Solutions larger than supremal normal) 5/18 X.Yin & S.Lafortune (UMich) IFAC World Congress 2014 August 25, 2014

Literature Survey • Heymann, Michael, and Feng Lin. "On-line control of partially observed discrete event systems." Discrete Event Dynamic Systems 4.3 (1994): 221-236. • Hadj-Alouane, Nejib Ben, Stéphane Lafortune, and Feng 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. (Online control; Only for safety specification; A certain class of maximal policies) 6/18 X.Yin & S.Lafortune (UMich) IFAC World Congress 2014 August 25, 2014

Literature Survey • Heymann, Michael, and Feng Lin. "On-line control of partially observed discrete event systems." Discrete Event Dynamic Systems 4.3 (1994): 221-236. • Hadj-Alouane, Nejib Ben, Stéphane Lafortune, and Feng 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. (Online control; Only for safety specification; A certain class of maximal policies) • K. Inan , “Nondeterministic supervision under partial observations,” in 11th International Conference on Analysis and Optimization of Systems: Discrete Event Systems. Springer, (1994): 39 – 48. (Decidability for safe and non-blocking; No synthesis) • T.-S. Yoo, and S. Lafortune. "Solvability of centralized supervisory control under partial observation." Discrete Event Dynamic Systems 16.4 (2006): 527-553. (Solvability for safe and non-blocking; No maximality) 6/18 X.Yin & S.Lafortune (UMich) IFAC World Congress 2014 August 25, 2014

The Need for a New Approach Why we need a new approach?  Observability is not preserved under union  algebraic approach cannot obtain a maximal solution  synthesis of maximally-permissive safe and non-blocking supervisor is open  Solution space may be infinite  how to solve optimal control problem? 7/18 X.Yin & S.Lafortune (UMich) IFAC World Congress 2014 August 25, 2014

Bipartite Transition System: A New Approach What is our new approach?  Bipartite transition system  A game structure between the controller and the system  Enumerates all (infinite) legal solutions using a finite structure  A state-based approach for synthesis  Inspired by methodologies in reactive synthesis literature 8/18 X.Yin & S.Lafortune (UMich) IFAC World Congress 2014 August 25, 2014

Bipartite Transition System Information State: a set of states, 𝐽 ≔ 2 𝑌 9/18 X.Yin & S.Lafortune (UMich) IFAC World Congress 2014 August 25, 2014

Bipartite Transition System Information State: a set of states, 𝐽 ≔ 2 𝑌 Definition. (BTS). A bipartite transition system T w.r.t. G is a 7-tuple 𝑈 = (𝑅 𝑍 , 𝑅 𝑎 , ℎ 𝑍𝑎 , ℎ 𝑎𝑍 , 𝐹, Γ, 𝑧 0 ) where • 𝑅 𝑍 ⊆ 𝐽 is the set of Y-states; • 𝑅 𝑎 ⊆ 𝐽 × Γ is the set of Z-states so that z = (𝐽 𝑨 , Γ 𝑨 ) ; • ℎ 𝑍𝑎 : 𝑅 𝑍 × Γ → Q 𝑎 represents the unobservable reach; 𝑈 × E → Q 𝑍 represents the observation transition; • ℎ 𝑎𝑍 : 𝑅 𝑎 • E is the set of events of G ; • Γ is the set of admissible control decisions of G ; • 𝑧 0 = {𝑦 0 } is the initial state. 9/18 X.Yin & S.Lafortune (UMich) IFAC World Congress 2014 August 25, 2014

Bipartite Transition System {0} 𝑐 1 𝑐 2 0 𝑝 1 𝑝 2 𝑝 2 𝑝 1 1 2 𝑝 1 𝑝 2 𝑝 2 𝑝 1 𝑑 1 𝑑 2 𝑑 1 𝑑 2 3 7 5 11 13 6 4 9 𝑑 2 𝑑 1 𝑑 2 𝑝 2 𝑑 1 𝑝 1 𝑝 2 𝑝 1 8 12 14 10 𝑑 2 𝑑 1 𝑑 1 𝑑 2 15 𝐹 𝑑 = {𝑑 1 , 𝑑 2 }, 𝐹 𝑝 = {𝑝 1 , 𝑝 2 } Illegal states = 𝑌 ∖ 𝑌 𝐼 = {15} 10/18 X.Yin & S.Lafortune (UMich) IFAC World Congress 2014 August 25, 2014

Bipartite Transition System {0} 𝑐 1 𝑐 2 { } 0 𝑝 1 𝑝 2 𝑝 2 𝑝 1 1 2 𝑝 1 𝑝 2 𝑝 2 𝑝 1 𝑑 1 𝑑 2 𝑑 1 𝑑 2 3 7 5 11 13 6 4 9 𝑑 2 𝑑 1 𝑑 2 𝑝 2 𝑑 1 𝑝 1 𝑝 2 𝑝 1 8 12 14 10 𝑑 2 𝑑 1 𝑑 1 𝑑 2 15 𝐹 𝑑 = {𝑑 1 , 𝑑 2 }, 𝐹 𝑝 = {𝑝 1 , 𝑝 2 } Illegal states = 𝑌 ∖ 𝑌 𝐼 = {15} 10/18 X.Yin & S.Lafortune (UMich) IFAC World Congress 2014 August 25, 2014

Bipartite Transition System {0} 𝑐 1 𝑐 2 { } 0 {0,1,2},{ } 𝑝 1 𝑝 2 𝑝 2 𝑝 1 1 2 𝑝 1 𝑝 2 𝑝 2 𝑝 1 𝑑 1 𝑑 2 𝑑 1 𝑑 2 3 7 5 11 13 6 4 9 𝑑 2 𝑑 1 𝑑 2 𝑝 2 𝑑 1 𝑝 1 𝑝 2 𝑝 1 8 12 14 10 𝑑 2 𝑑 1 𝑑 1 𝑑 2 15 𝐹 𝑑 = {𝑑 1 , 𝑑 2 }, 𝐹 𝑝 = {𝑝 1 , 𝑝 2 } Illegal states = 𝑌 ∖ 𝑌 𝐼 = {15} 10/18 X.Yin & S.Lafortune (UMich) IFAC World Congress 2014 August 25, 2014

Bipartite Transition System {0} 𝑐 1 𝑐 2 { } 0 𝑝 1 {0,1,2},{ } 𝑝 1 𝑝 2 𝑝 2 𝑝 1 1 2 𝑝 1 𝑝 2 𝑝 2 𝑝 1 𝑑 1 𝑑 2 𝑑 1 𝑑 2 3 7 5 11 13 6 4 9 𝑑 2 𝑑 1 𝑑 2 𝑝 2 𝑑 1 𝑝 1 𝑝 2 𝑝 1 8 12 14 10 𝑑 2 𝑑 1 𝑑 1 𝑑 2 15 𝐹 𝑑 = {𝑑 1 , 𝑑 2 }, 𝐹 𝑝 = {𝑝 1 , 𝑝 2 } Illegal states = 𝑌 ∖ 𝑌 𝐼 = {15} 10/18 X.Yin & S.Lafortune (UMich) IFAC World Congress 2014 August 25, 2014

Bipartite Transition System {0} 𝑐 1 𝑐 2 { } 0 𝑝 1 {0,1,2},{ } {3,4} 𝑝 1 𝑝 2 𝑝 2 𝑝 1 1 2 𝑝 1 𝑝 2 𝑝 2 𝑝 1 𝑑 1 𝑑 2 𝑑 1 𝑑 2 3 7 5 11 13 6 4 9 𝑑 2 𝑑 1 𝑑 2 𝑝 2 𝑑 1 𝑝 1 𝑝 2 𝑝 1 8 12 14 10 𝑑 2 𝑑 1 𝑑 1 𝑑 2 15 𝐹 𝑑 = {𝑑 1 , 𝑑 2 }, 𝐹 𝑝 = {𝑝 1 , 𝑝 2 } Illegal states = 𝑌 ∖ 𝑌 𝐼 = {15} 10/18 X.Yin & S.Lafortune (UMich) IFAC World Congress 2014 August 25, 2014