minimization of sensor activation in decentralized fault diagnosis of discrete … ·...
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Xiang Yin and Stéphane Lafortune
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Minimization of Sensor Activation in Decentralized Fault Diagnosis of Discrete Event Systems
EECS Department, University of Michigan
54th IEEE CDC, Dec 15-18, 2015, Osaka, Japan
X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
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Introduction
X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
𝑃2(𝑠)
𝑃2
𝑠
𝑃1(𝑠)
𝑃1
𝑠
𝑠
𝐷1 𝐷2
Coordinator
Fault Alarm
Plant G
0
1
2 3
4
5
Agent 1 Agent 2
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Introduction
X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
𝑃Ω2(𝑠)
𝑃2
𝑠
Ω2
𝑃Ω1(𝑠)
𝑃1
𝑠
Ω1
𝑠
𝐷1 𝐷2
Coordinator
Fault Alarm
Plant G
0
1
2 3
4
5
Agent 1 Agent 2
System Model
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𝐺 = (𝑄, Σ, 𝛿, 𝑞0) is a deterministic FSA
• 𝑄 is the finite set of states; • Σ is the finite set of events; • 𝛿: 𝑄 × Σ → 𝑄 is the partial transition function; • 𝑞0 is the initial state.
System Model
- Sensor activation policy Ω = (𝐴, 𝐿), where 𝐴 = (𝑄𝐴, Σ𝑜, 𝛿𝐴, 𝑞0,𝐴) and 𝐿: 𝑄𝐴 → 2Σ𝑜;
- Projection 𝑃Ω: ℒ 𝐺 → Σ𝑜∗
- State estimate ℰΩ𝐺 𝑠
- Observer 𝑂𝑏𝑠Ω 𝐺 = 𝑋, Σ𝑜, 𝑓, 𝑥0 , and 𝑥 = 𝐼 𝑥 , 𝐴 𝑥 , 𝐼 𝑥 ∈ 2𝑄. 𝐴 𝑥 ∈ 𝑄𝐴
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𝐺 = (𝑄, Σ, 𝛿, 𝑞0) is a deterministic FSA
• 𝑄 is the finite set of states; • Σ is the finite set of events; • 𝛿: 𝑄 × Σ → 𝑄 is the partial transition function; • 𝑞0 is the initial state.
1
5 4
6
2 3
𝑓
7
𝑓
𝑜 𝑜
𝑜
𝑏
𝑏 𝑎
𝑎
𝑜
2 3 𝑜 𝑎
1 *𝑜+ *𝑎+ ∅
( 2,4,7 , 2)
𝑜
𝑎 ( 6 , 3)
( 1,3,5,7 , 1)
𝛀 𝑶𝒃𝒔𝜴 𝑮 𝑮
Decentralized Diagnosis Problem
• A fault event 𝑒𝑑 ∈ Σ ∖ (∪𝑖=1,2 Σ𝑜,𝑖)
• Ψ 𝑒𝑑 = *𝑠𝑒𝑑 ∈ ℒ 𝐺 : 𝑠 ∈ Σ∗+
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• Two agents ℐ = *1,2+, Ω = Ω1, Ω2 with Σ𝑜,1 and Σ𝑜,2
Decentralized Diagnosis Problem
• 𝑋 is the finite set of states; • 𝐸 is the finite set of events; • 𝑓: 𝑋 × 𝐸 → 𝑋 is the partial transition function; • 𝑋0 is the set of initial states.
• K-Codiagnosability:
A live language ℒ 𝐺 is said to be 𝐾-codiagnosable w.r.t. Ω and 𝑒𝑑 if
(∀𝑠 ∈ Ψ(𝑒𝑑 ))(∀𝑡 ∈ ℒ 𝐺 /𝑠), 𝑡 ≥ 𝐾 ⇒ 𝐶𝐷-
where the codiagnosability condition 𝐶𝐷 is
∃𝑖 ∈ *1,2+ ∀𝜔 ∈ ℒ 𝐺 𝑃Ω𝑖𝑤 = 𝑃Ω𝑖
𝑠𝑡 ⇒ 𝑒𝑑 ∈ 𝜔 .
• A fault event 𝑒𝑑 ∈ Σ ∖ (∪𝑖=1,2 Σ𝑜,𝑖)
• Ψ 𝑒𝑑 = *𝑠𝑒𝑑 ∈ ℒ 𝐺 : 𝑠 ∈ Σ∗+
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• Two agents ℐ = *1,2+, Ω = Ω1, Ω2 with Σ𝑜,1 and Σ𝑜,2
Problem Formulation
• Ω′< Ω
∗ is defined in terms of set inclusion.
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• 𝑋 is the finite set of states; • 𝐸 is the finite set of events; • 𝑓: 𝑋 × 𝐸 → 𝑋 is the partial transition function; • 𝑋0 is the set of initial states.
• Decentralized Minimization Problem
Let 𝐺 be the system with fault event 𝑒𝑑. For each agent 𝑖 ∈ 1,2 , let Σ𝑜,𝑖 ⊆ Σ be the set of observable events. Find a sensor activation policy
Ω∗= ,Ω1
∗ , Ω2∗ - such that
C1. ℒ 𝐺 is 𝐾-codiagnosable w.r.t. Ω∗ and ed;
C2. Ω∗ is minimal, i.e., there does not exist another Ω
′< Ω
∗ that satisfies (C1).
Literature Review
Decentralized Fault Diagnosis • Debouk, R., Lafortune, S., & Teneketzis, D. (2000). Coordinated decentralized protocols for failure diagnosis of discrete
event systems. Discrete Event Dynamic Systems, 10(1-2), 33-86. • Qiu, W., & Kumar, R. (2006). Decentralized failure diagnosis of discrete event systems. IEEE Transactions on Systems,
Man and Cybernetics, Part A: Systems and Humans, 36(2), 384-395. • Kumar, R., & Takai, S. (2009). Inference-based ambiguity management in decentralized decision-making: Decentralized
diagnosis of discrete-event systems. IEEE Transactions on Automation Science and Engineering, 6(3), 479-491. • Moreira, M. V., Jesus, T. C., & Basilio, J. C. (2011). Polynomial time verification of decentralized diagnosability of
discrete event systems. IEEE Transactions on Automatic Control, 56(7), 1679-1684.
Dynamic Sensor Activation Problem • Thorsley, D., & Teneketzis, D. (2007). Active acquisition of information for diagnosis and supervisory control of discrete
event systems. Discrete Event Dynamic Systems, 17(4), 531-583. • Cassez, F., & Tripakis, S. (2008). Fault diagnosis with static and dynamic observers. Fundamenta Informaticae, 88(4),
497-540. • Cassez, F., Dubreil, J., & Marchand, H. (2012). Synthesis of opaque systems with static and dynamic masks. Formal
Methods in System Design, 40(1), 88-115. • Shu, S., Huang, Z., & Lin, F. (2013). Online sensor activation for detectability of discrete event systems. IEEE
Transactions on Automation Science and Engineering, 10(2), 457-461. • Wang, W., Lafortune, S., Lin, F., & Girard, A. R. (2010). Minimization of dynamic sensor activation in discrete event
systems for the purpose of control. IEEE Transactions on Automatic Control, 55(11), 2447-2461. • Wang, W., Lafortune, S., Girard, A. R., & Lin, F. (2010). Optimal sensor activation for diagnosing discrete event systems.
Automatica, 46(7), 1165-1175.
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Solution Overview
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𝛀𝟏𝟎 𝛀𝟐
𝟎
Person by Person Approach
Agent 1 Agent 2
Solution Overview
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𝛀𝟏𝟎 𝛀𝟐
𝟎
𝛀𝟏𝟎
Person by Person Approach
Agent 1 Agent 2
Solution Overview
6/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
𝛀𝟏𝟎 𝛀𝟐
𝟎
𝛀𝟏𝟎 𝛀𝟐
𝟏
Person by Person Approach
Agent 1 Agent 2
Solution Overview
6/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
𝛀𝟏𝟎 𝛀𝟐
𝟎
𝛀𝟏𝟎 𝛀𝟐
𝟏
𝛀𝟐𝟏
Person by Person Approach
Agent 1 Agent 2
Solution Overview
6/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
𝛀𝟏𝟎 𝛀𝟐
𝟎
𝛀𝟏𝟎 𝛀𝟐
𝟏
𝛀𝟏𝟏 𝛀𝟐
𝟏
Person by Person Approach
Agent 1 Agent 2
Solution Overview
6/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
𝛀𝟏𝟎 𝛀𝟐
𝟎
𝛀𝟏𝟎 𝛀𝟐
𝟏
𝛀𝟏𝟏 𝛀𝟐
𝟏
Person by Person Approach
Agent 1 Agent 2
Solution Overview
6/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
𝛀𝟏𝟎 𝛀𝟐
𝟎
𝛀𝟏𝟎 𝛀𝟐
𝟏
𝛀𝟏𝟏 𝛀𝟐
𝟏
𝛀𝟏∗ 𝛀𝟐
∗
Person by Person Approach
Agent 1 Agent 2
Solution Overview
6/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
𝛀𝟏𝟎 𝛀𝟐
𝟎
𝛀𝟏𝟎 𝛀𝟐
𝟏
𝛀𝟏𝟏 𝛀𝟐
𝟏
𝛀𝟏∗ 𝛀𝟐
∗
Challenges & Solutions
• Constrained minimization problem
Person by Person Approach
Agent 1 Agent 2
Solution Overview
6/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
𝛀𝟏𝟎 𝛀𝟐
𝟎
𝛀𝟏𝟎 𝛀𝟐
𝟏
𝛀𝟏𝟏 𝛀𝟐
𝟏
𝛀𝟏∗ 𝛀𝟐
∗
Challenges & Solutions
• Constrained minimization problem
Person by Person Approach
Agent 1 Agent 2
- Full centralized problem - Generalized state-partition automaton
Solution Overview
6/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
𝛀𝟏𝟎 𝛀𝟐
𝟎
𝛀𝟏𝟎 𝛀𝟐
𝟏
𝛀𝟏𝟏 𝛀𝟐
𝟏
𝛀𝟏∗ 𝛀𝟐
∗
Challenges & Solutions
• Constrained minimization problem
• Converge?
Person by Person Approach
Agent 1 Agent 2
- Full centralized problem - Generalized state-partition automaton
Solution Overview
6/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
𝛀𝟏𝟎 𝛀𝟐
𝟎
𝛀𝟏𝟎 𝛀𝟐
𝟏
𝛀𝟏𝟏 𝛀𝟐
𝟏
𝛀𝟏∗ 𝛀𝟐
∗
Challenges & Solutions
• Constrained minimization problem
• Converge?
Person by Person Approach
Agent 1 Agent 2
- Full centralized problem - Generalized state-partition automaton
- Yes! - Monotonicity property
Solution Overview
6/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
𝛀𝟏𝟎 𝛀𝟐
𝟎
𝛀𝟏𝟎 𝛀𝟐
𝟏
𝛀𝟏𝟏 𝛀𝟐
𝟏
𝛀𝟏∗ 𝛀𝟐
∗
Challenges & Solutions
• Constrained minimization problem
• Converge?
• Minimal?
Person by Person Approach
Agent 1 Agent 2
- Full centralized problem - Generalized state-partition automaton
- Yes! - Monotonicity property
Solution Overview
6/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
𝛀𝟏𝟎 𝛀𝟐
𝟎
𝛀𝟏𝟎 𝛀𝟐
𝟏
𝛀𝟏𝟏 𝛀𝟐
𝟏
𝛀𝟏∗ 𝛀𝟐
∗
Challenges & Solutions
• Constrained minimization problem
• Converge?
• Minimal?
Person by Person Approach
Agent 1 Agent 2
- Full centralized problem - Generalized state-partition automaton
- Yes! - Monotonicity property
- Yes! - Logical optimal (set inclusion)
Generalized State-Partition Automaton
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• Generalized State-Partition Automaton
Let 𝐺 be an automaton, Ω a sensor activation policy and 𝑂𝑏𝑠Ω 𝐺 be the corresponding observer. We sat that 𝐺 is a state-partition automaton (SPA) w.r.t. Ω, if
∀𝑥, 𝑦 ∈ 𝑋: 𝐼 𝑥 = 𝐼 𝑦 or 𝐼 𝑥 ∩ 𝐼 𝑦 ≠ ∅
Generalized State-Partition Automaton
7/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
• Generalized State-Partition Automaton
Let 𝐺 be an automaton, Ω a sensor activation policy and 𝑂𝑏𝑠Ω 𝐺 be the corresponding observer. We sat that 𝐺 is a state-partition automaton (SPA) w.r.t. Ω, if
∀𝑥, 𝑦 ∈ 𝑋: 𝐼 𝑥 = 𝐼 𝑦 or 𝐼 𝑥 ∩ 𝐼 𝑦 ≠ ∅
Cho, H., & Marcus, S. I. (1989). On supremal languages of classes of sublanguages that arise in supervisor synthesis problems with partial observation. Mathematics of Control, Signals and Systems, 2(1), 47-69.
• SPA for Static Observations
Generalized State-Partition Automaton
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• Generalized State-Partition Automaton
Let 𝐺 be an automaton, Ω a sensor activation policy and 𝑂𝑏𝑠Ω 𝐺 be the corresponding observer. We sat that 𝐺 is a state-partition automaton (SPA) w.r.t. Ω, if
∀𝑥, 𝑦 ∈ 𝑋: 𝐼 𝑥 = 𝐼 𝑦 or 𝐼 𝑥 ∩ 𝐼 𝑦 ≠ ∅
( 7 , 1)
𝑏
𝑏
( 5 , 1)
( 1,2,3,4,6 , 1) 1
5 4
6
2 3
𝑓
7
𝑓
𝑜
𝑜
𝑏
𝑏 𝑎
𝑎
𝑜
1 𝑏 *𝑏+
Ω1
Generalized State-Partition Automaton
7/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
• Generalized State-Partition Automaton
Let 𝐺 be an automaton, Ω a sensor activation policy and 𝑂𝑏𝑠Ω 𝐺 be the corresponding observer. We sat that 𝐺 is a state-partition automaton (SPA) w.r.t. Ω, if
∀𝑥, 𝑦 ∈ 𝑋: 𝐼 𝑥 = 𝐼 𝑦 or 𝐼 𝑥 ∩ 𝐼 𝑦 ≠ ∅
( 7 , 1)
𝑏
𝑏
( 5 , 1)
( 1,2,3,4,6 , 1) 1
5 4
6
2 3
𝑓
7
𝑓
𝑜 𝑜
𝑜
𝑏
𝑏 𝑎
𝑎
𝑜
1 𝑏 *𝑏+
2 3 𝑜 𝑎
1 *𝑜+ *𝑎+ ∅ ( 2,4, 𝟕 , 2)
𝑜
𝑎 ( 6 , 3)
( 1,3,5, 𝟕 , 1) Ω1 Ω2
Generalized State-Partition Automaton
7/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
• Generalized State-Partition Automaton
Let 𝐺 be an automaton, Ω a sensor activation policy and 𝑂𝑏𝑠Ω 𝐺 be the corresponding observer. We sat that 𝐺 is a state-partition automaton (SPA) w.r.t. Ω, if
∀𝑥, 𝑦 ∈ 𝑋: 𝐼 𝑥 = 𝐼 𝑦 or 𝐼 𝑥 ∩ 𝐼 𝑦 ≠ ∅
( 7 , 1)
𝑏
𝑏
( 5 , 1)
( 1,2,3,4,6 , 1) 1
5 4
6
2 3
𝑓
7
𝑓
𝑜 𝑜
𝑜
𝑏
𝑏 𝑎
𝑎
𝑜
1 𝑏 *𝑏+
2 3 𝑜 𝑎
1 *𝑜+ *𝑎+ ∅ ( 2,4,7 , 2)
𝑜
𝑎 ( 6 , 3)
( 1,3,5,7 , 1)
• Theorem
Let 𝐺 be the system automaton, Ω be a sensor activation policy. Then 𝑂𝑏𝑠Ω
+ 𝐺 ∥ 𝐺 is an SPA w.r.t. Ω such that ℒ 𝑂𝑏𝑠Ω+ 𝐺 ∥ 𝐺 = ℒ 𝐺 .
Ω1 Ω2
Inference Function
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Suppose that 𝐺 is an SPA w.r.t. 𝛺 and 𝑂𝑏𝑠𝛺 𝐺 = (𝑋, Σ𝑜, 𝑓, 𝑥0) is the observer. Then for any state q ∈ 𝑄, there exists a unique information state ℱ 𝑞 ∈ 2𝑄 s.t.
𝑞 ∈ ℱ 𝑞 and ∃𝑞𝐴 ∈ 𝑄𝐴: ℱ 𝑞 , 𝑞𝐴 ∈ 𝑋
We call this information state ℱ 𝑞 the inference of state 𝑞. ℱ:𝑄 → 2𝑄 such that
∀𝑠 ∈ ℒ 𝐺 : 𝛿 𝑞0, 𝑠 = 𝑞 ⇒ ,ℱ 𝑞 = 𝐼(𝑓(𝑃Ω(𝑠)))-
( 7 , 1)
𝑏
𝑏
( 5 , 1)
( 1,2,3,4,6 , 1) 1
5 4
6
2 3
𝑓
7
𝑓
𝑜 𝑜
𝑜
𝑏
𝑏 𝑎
𝑎
𝑜 1 𝑏 *𝑏+
• Inference Function
Inference Function
8/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
Suppose that 𝐺 is an SPA w.r.t. 𝛺 and 𝑂𝑏𝑠𝛺 𝐺 = (𝑋, Σ𝑜, 𝑓, 𝑥0) is the observer. Then for any state q ∈ 𝑄, there exists a unique information state ℱ 𝑞 ∈ 2𝑄 s.t.
𝑞 ∈ ℱ 𝑞 and ∃𝑞𝐴 ∈ 𝑄𝐴: ℱ 𝑞 , 𝑞𝐴 ∈ 𝑋
We call this information state ℱ 𝑞 the inference of state 𝑞. ℱ:𝑄 → 2𝑄 such that
∀𝑠 ∈ ℒ 𝐺 : 𝛿 𝑞0, 𝑠 = 𝑞 ⇒ ,ℱ 𝑞 = 𝐼(𝑓(𝑃Ω(𝑠)))-
𝑏
𝑏
1
5 4
6
2 3
𝑓
7
𝑓
𝑜 𝑜
𝑜
𝑏
𝑏 𝑎
𝑎
𝑜 1 𝑏 *𝑏+
• Inference Function
( 1,2,3,4,6 , 1)
( 5 , 1)
( 7 , 1)
Problem Reformulation
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𝐺 = (𝑄 , Σ, 𝛿 , 𝑞 0) is a deterministic FSA
• 𝑄 = 𝑄 × −1,0,1, … , 𝐾 and 𝑞 0 = 𝑞0, −1 .
K-Augmented Automaton
1
5 4
6
2 3
𝑓
7
𝑓
𝑜 𝑜
𝑜
𝑏
𝑏 𝑎
𝑎
𝑜
𝑓
𝑓
𝑜 𝑜
𝑜
𝑏
𝑏 𝑎
𝑎
𝑜 2 , - 1 1 , - 1
4 , 0
6 , 1
3 , 0
5 , 1
7 , 1
𝑮 𝑮
𝑲 = 𝟏
Problem Reformulation
9/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
𝐺 = (𝑄 , Σ, 𝛿 , 𝑞 0) is a deterministic FSA
• 𝑄 = 𝑄 × −1,0,1, … , 𝐾 and 𝑞 0 = 𝑞0, −1 .
K-Augmented Automaton
• 𝐷𝐼 𝑥 = 𝑁 if ∀𝑞 ∈ 𝑥: 𝑞 𝑛 = −1
• 𝐷𝐼 𝑥 = 𝐹 if ∀𝑞 ∈ 𝑥: 𝑞 𝑛 ≥ 0
• 𝐷𝐼 𝑥 = 𝐶1 if ,∀𝑞 ∈ 𝑥: 𝑞 𝑛 ≠ 𝐾- ∧ ,∃𝑞, 𝑞′ ∈ 𝑥: 𝑞 𝑛 = −1 ∧ 0 ≤ 𝑞′ 𝑛 < 𝐾-
• 𝐷𝐼 𝑥 = 𝐶2 if ∃𝑞, 𝑞′ ∈ 𝑥: 𝑞 𝑛 = −1 ∧ 𝑞′ 𝑛 = 𝐾
Diagnosability Function
Centralized Constrained Minimization Problem
• 𝑋 is the finite set of states; • 𝐸 is the finite set of events; • 𝑓: 𝑋 × 𝐸 → 𝑋 is the partial transition function; • 𝑋0 is the set of initial states.
• Centralized Constrained Minimization Problem
Let 𝑖, 𝑗 ∈ 1,2 , 𝑖 ≠ 𝑗 be two agent. Suppose that the sensor activation policy Ω𝑗
for Agent 𝑗 is fixed. Find a sensor activation policy Ω𝑖 for Agent 𝑖
s.t. C1. ℒ 𝐺 is 𝐾-codiagnosable w.r.t. Ω1
, Ω2 ;
C2. For any Ω𝑖′ satisfying (C1), we have Ω𝑖
′ ≮ Ω𝑖
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Centralized Constrained Minimization Problem
• 𝑋 is the finite set of states; • 𝐸 is the finite set of events; • 𝑓: 𝑋 × 𝐸 → 𝑋 is the partial transition function; • 𝑋0 is the set of initial states.
• Centralized Constrained Minimization Problem
Let 𝑖, 𝑗 ∈ 1,2 , 𝑖 ≠ 𝑗 be two agent. Suppose that the sensor activation policy Ω𝑗
for Agent 𝑗 is fixed. Find a sensor activation policy Ω𝑖 for Agent 𝑖
s.t. C1. ℒ 𝐺 is 𝐾-codiagnosable w.r.t. Ω1
, Ω2 ;
C2. For any Ω𝑖′ satisfying (C1), we have Ω𝑖
′ ≮ Ω𝑖
10/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
• 𝑋 is the finite set of states; • 𝐸 is the finite set of events; • 𝑓: 𝑋 × 𝐸 → 𝑋 is the partial transition function; • 𝑋0 is the set of initial states.
• Centralized Sensor Minimization Problem for IS-Based Property
Let 𝐺 = (𝑄, Σ, 𝛿, 𝑞0) be the system and 𝜙: 2𝑄 → *0,1+ be a function on information states. Find a sensor activation policy Ω s.t.
C1. ∀𝑠 ∈ ℒ 𝐺 :𝜙 ℰΩ𝐺 𝑠 = 1;
C2. For any Ω′ satisfying (C1), we have Ω′ ≮ Ω .
Problem Reduction
11/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
𝐶𝐷𝑖 𝑥 = 0, if
𝐷𝐼 𝑥 = 𝐶2 𝑎𝑛𝑑
(∃𝑞 ∈ 𝑥), 𝑞 𝑛 = 𝐾 ∧ 𝐷𝐼(ℱ𝑗(𝑞)) ≠ 𝐹-
1, otherwise
Suppose that 𝐺 = 𝑄 , Σ, 𝛿 , 𝑞 0 is a SPA w.r.t. Ω𝑗 and ℱ𝑗: 2
𝑄 → *0,1+ is the
corresponding inference function. We define the codiagnosability function
𝐶𝐷𝑖 𝑥 : 2𝑄 → *0,1+ for Agent i as follows. For each 𝑥 ∈ 2𝑄
Problem Reduction
11/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
𝐶𝐷𝑖 𝑥 = 0, if
𝐷𝐼 𝑥 = 𝐶2 𝑎𝑛𝑑
(∃𝑞 ∈ 𝑥), 𝑞 𝑛 = 𝐾 ∧ 𝐷𝐼(ℱ𝑗(𝑞)) ≠ 𝐹-
1, otherwise
Suppose that 𝐺 = 𝑄 , Σ, 𝛿 , 𝑞 0 is a SPA w.r.t. Ω𝑗 and ℱ𝑗: 2
𝑄 → *0,1+ is the
corresponding inference function. We define the codiagnosability function
𝐶𝐷𝑖 𝑥 : 2𝑄 → *0,1+ for Agent i as follows. For each 𝑥 ∈ 2𝑄
-1 K
𝑃Ω𝑖 𝑠 = 𝑃Ω𝑖(𝑡)
𝑡
Agent i
𝑠
Problem Reduction
11/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
𝐶𝐷𝑖 𝑥 = 0, if
𝐷𝐼 𝑥 = 𝐶2 𝑎𝑛𝑑
(∃𝑞 ∈ 𝑥), 𝑞 𝑛 = 𝐾 ∧ 𝐷𝐼(ℱ𝑗(𝑞)) ≠ 𝐹-
1, otherwise
Suppose that 𝐺 = 𝑄 , Σ, 𝛿 , 𝑞 0 is a SPA w.r.t. Ω𝑗 and ℱ𝑗: 2
𝑄 → *0,1+ is the
corresponding inference function. We define the codiagnosability function
𝐶𝐷𝑖 𝑥 : 2𝑄 → *0,1+ for Agent i as follows. For each 𝑥 ∈ 2𝑄
-1 K
𝑃Ω𝑖 𝑠 = 𝑃Ω𝑖(𝑡)
𝑡
𝓕𝒋 K Agent j
Agent i
-1
𝑠
Problem Reduction
11/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
𝐶𝐷𝑖 𝑥 = 0, if
𝐷𝐼 𝑥 = 𝐶2 𝑎𝑛𝑑
(∃𝑞 ∈ 𝑥), 𝑞 𝑛 = 𝐾 ∧ 𝐷𝐼(ℱ𝑗(𝑞)) ≠ 𝐹-
1, otherwise
Suppose that 𝐺 = 𝑄 , Σ, 𝛿 , 𝑞 0 is a SPA w.r.t. Ω𝑗 and ℱ𝑗: 2
𝑄 → *0,1+ is the
corresponding inference function. We define the codiagnosability function
𝐶𝐷𝑖 𝑥 : 2𝑄 → *0,1+ for Agent i as follows. For each 𝑥 ∈ 2𝑄
-1 K
𝑃Ω𝑖 𝑠 = 𝑃Ω𝑖(𝑡)
𝑡
𝓕𝒋 K Agent j
Agent i
-1
𝑠
-1 K
Problem Reduction
11/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
𝐶𝐷𝑖 𝑥 = 0, if
𝐷𝐼 𝑥 = 𝐶2 𝑎𝑛𝑑
(∃𝑞 ∈ 𝑥), 𝑞 𝑛 = 𝐾 ∧ 𝐷𝐼(ℱ𝑗(𝑞)) ≠ 𝐹-
1, otherwise
Suppose that 𝐺 = 𝑄 , Σ, 𝛿 , 𝑞 0 is a SPA w.r.t. Ω𝑗 and ℱ𝑗: 2
𝑄 → *0,1+ is the
corresponding inference function. We define the codiagnosability function
𝐶𝐷𝑖 𝑥 : 2𝑄 → *0,1+ for Agent i as follows. For each 𝑥 ∈ 2𝑄
𝑠 𝑃Ω𝑖 𝑠 = 𝑃Ω𝑖
(𝑡) 𝑡
𝓕𝒋 K K-1
Agent j
Agent i
Problem Reduction
12/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
• Theorem.
Suppose that 𝐺 = 𝑄 , Σ, 𝛿 , 𝑞 0 is a SPA w.r.t. Ω𝑗 . Then ℒ 𝐺 is 𝐾-codiagnosable
w.r.t. Ω1 , Ω2 and 𝑒𝑑, if and only if,
∀𝑠 ∈ ℒ 𝐺 : 𝐶𝐷𝑖 ℰΩ𝐺 𝑠 = 1
Problem Reduction
12/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
• Theorem.
Suppose that 𝐺 = 𝑄 , Σ, 𝛿 , 𝑞 0 is a SPA w.r.t. Ω𝑗 . Then ℒ 𝐺 is 𝐾-codiagnosable
w.r.t. Ω1 , Ω2 and 𝑒𝑑, if and only if,
∀𝑠 ∈ ℒ 𝐺 : 𝐶𝐷𝑖 ℰΩ𝐺 𝑠 = 1
• Centralized Sensor Minimization Problem for IS-Based Property
Let 𝐺 = (𝑄, Σ, 𝛿, 𝑞0) be the system and 𝜙: 2𝑄 → *0,1+ be a function on information states. Find a sensor activation policy Ω s.t.
C1. ∀𝑠 ∈ ℒ 𝐺 :𝜙(ℰΩ𝐺 𝑠 ) = 1
Problem Reduction
12/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
X. Yin and S. Lafortune, “A General Approach for Solving Dynamic Sensor Activation Problems for a Class of Properties”
Wednesday December 16, 17:20-17:40, Switched Systems III, WeC10
• Theorem
The centralized constrained minimization problem can be effectively solve.
• Theorem.
Suppose that 𝐺 = 𝑄 , Σ, 𝛿 , 𝑞 0 is a SPA w.r.t. Ω𝑗 . Then ℒ 𝐺 is 𝐾-codiagnosable
w.r.t. Ω1 , Ω2 and 𝑒𝑑, if and only if,
∀𝑠 ∈ ℒ 𝐺 : 𝐶𝐷𝑖 ℰΩ𝐺 𝑠 = 1
Synthesis Algorithm
13/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
𝑓
𝑓
𝑜
𝑜
𝑏
𝑏 𝑎
𝑎
𝑜 2 , - 1 1 , - 1
4 , 0
6 , 1
3 , 0
5 , 1
7 , 1
1 𝑏 *𝑏+
1 𝑎, 𝑜 *𝑎, 𝑜+
𝑜
Agent 1:𝚺𝒐,𝟏 = *𝒃+ Agent 2:𝚺𝒐,𝟏 = *𝒐, 𝒂+
Synthesis Algorithm
13/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
𝑓
𝑓
𝑜
𝑜
𝑏
𝑏 𝑎
𝑎
𝑜 2 , - 1 1 , - 1
4 , 0
6 , 1
3 , 0
5 , 1
7 , 1
1 𝑏 *𝑏+
1 𝑎, 𝑜 *𝑎, 𝑜+
1 𝑏 *𝑏+
𝑜
Agent 1:𝚺𝒐,𝟏 = *𝒃+ Agent 2:𝚺𝒐,𝟏 = *𝒐, 𝒂+
Synthesis Algorithm
13/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
𝑓
𝑓
𝑜
𝑜
𝑏
𝑏 𝑎
𝑎
𝑜 2 , - 1 1 , - 1
4 , 0
6 , 1
3 , 0
5 , 1
7 , 1
1 𝑏 *𝑏+
2 3 𝑜 𝑎
1 *𝑜+ *𝑎+ ∅
1 𝑎, 𝑜 *𝑎, 𝑜+
1 𝑏 *𝑏+
𝑜
Agent 1:𝚺𝒐,𝟏 = *𝒃+ Agent 2:𝚺𝒐,𝟏 = *𝒐, 𝒂+
Synthesis Algorithm
13/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
𝑓
𝑓
𝑜
𝑜
𝑏
𝑏 𝑎
𝑎
𝑜 2 , - 1 1 , - 1
4 , 0
6 , 1
3 , 0
5 , 1
7 , 1
1 𝑏 *𝑏+
2 3 𝑜 𝑎
1 *𝑜+ *𝑎+ ∅
1 𝑎, 𝑜 *𝑎, 𝑜+
1 𝑏 *𝑏+
𝑜
2 3 𝑜 𝑎
1 *𝑜+ *𝑎+ ∅
Agent 1:𝚺𝒐,𝟏 = *𝒃+ Agent 2:𝚺𝒐,𝟏 = *𝒐, 𝒂+
Synthesis Algorithm
13/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
𝑓
𝑓
𝑜
𝑜
𝑏
𝑏 𝑎
𝑎
𝑜 2 , - 1 1 , - 1
4 , 0
6 , 1
3 , 0
5 , 1
7 , 1
1 𝑏 *𝑏+
2 3 𝑜 𝑎
1 *𝑜+ *𝑎+ ∅
( 2,4, 𝟕 , 2)
𝑜
𝑎 ( 6 , 3)
( 1,3,5, 𝟕 , 1)
1 𝑎, 𝑜 *𝑎, 𝑜+
1 𝑏 *𝑏+
𝑜
2 3 𝑜 𝑎
1 *𝑜+ *𝑎+ ∅
Agent 1:𝚺𝒐,𝟏 = *𝒃+ Agent 2:𝚺𝒐,𝟏 = *𝒐, 𝒂+
Synthesis Algorithm
13/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
𝑓
𝑓
𝑜
𝑜
𝑏
𝑏 𝑎
𝑎
𝑜 2 , - 1 1 , - 1
4 , 0
6 , 1
3 , 0
5 , 1
7 , 1
1 𝑏 *𝑏+
2 3 𝑜 𝑎
1 *𝑜+ *𝑎+ ∅
( 2,4,7′ , 2)
𝑜
𝑎 ( 6 , 3)
( 1,3,5,7 , 1)
1 𝑎, 𝑜 *𝑎, 𝑜+
1 𝑏 *𝑏+
𝑓
𝑓
𝑜
𝑜
𝑏
𝑏 𝑎
𝑎
𝑜 2 , - 1 1 , - 1
4 , 0
6 , 1
3 , 0
5 , 1
7 , 1 𝑜 𝑜 7’ , 1
𝑜
2 3 𝑜 𝑎
1 *𝑜+ *𝑎+ ∅
Agent 1:𝚺𝒐,𝟏 = *𝒃+ Agent 2:𝚺𝒐,𝟏 = *𝒐, 𝒂+
Synthesis Algorithm
13/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
𝑓
𝑓
𝑜
𝑜
𝑏
𝑏 𝑎
𝑎
𝑜 2 , - 1 1 , - 1
4 , 0
6 , 1
3 , 0
5 , 1
7 , 1
1 𝑏 *𝑏+
2 3 𝑜 𝑎
1 *𝑜+ *𝑎+ ∅
1 𝑎, 𝑜 *𝑎, 𝑜+
1 𝑏 *𝑏+
𝑜
2 3 𝑜 𝑎
1 *𝑜+ *𝑎+ ∅
2 𝑏
1 *𝑏+ ∅
Agent 1:𝚺𝒐,𝟏 = *𝒃+ Agent 2:𝚺𝒐,𝟏 = *𝒐, 𝒂+
Synthesis Algorithm
13/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
𝑓
𝑓
𝑜
𝑜
𝑏
𝑏 𝑎
𝑎
𝑜 2 , - 1 1 , - 1
4 , 0
6 , 1
3 , 0
5 , 1
7 , 1
1 𝑏 *𝑏+
2 3 𝑜 𝑎
1 *𝑜+ *𝑎+ ∅
1 𝑎, 𝑜 *𝑎, 𝑜+
1 𝑏 *𝑏+
𝑜
2 3 𝑜 𝑎
1 *𝑜+ *𝑎+ ∅
2 𝑏
1 *𝑏+ ∅
Agent 1:𝚺𝒐,𝟏 = *𝒃+ Agent 2:𝚺𝒐,𝟏 = *𝒐, 𝒂+
Correctness
14/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
• Theorem.
Let Ω ∗ be the output of Algorithm D-MIN-ACT. Then Ω ∗ is a minimal solution.
Correctness
14/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
• Theorem.
Let Ω ∗ be the output of Algorithm D-MIN-ACT. Then Ω ∗ is a minimal solution.
Sketch of the Proof: • Monotonicity Property [Wang et al. 2011]. • Suppose that Ω ′ ≤ Ω ℒ G is K-codiagnosable w.r.t. Ω ′ implies that ℒ G is K-codiagnosable w.r.t. Ω .
Summary
15/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015
Contributions:
• A new person-by-person approach for synthesizing decentralized sensor activation policies for the purpose of fault diagnosis
• Generalized state-partition automaton for dynamic observations
• The solution is provably language-based minimal
• The approach that we proposed is also applicable to the problem of decentralized sensor activation for the purpose of control