Minimization of Sensor Activation in Decentralized Fault Diagnosis of Discrete … ·...

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

1/15

Introduction


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

1/15

Introduction


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

2/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

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


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∗+


• 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∗+


• 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.



• 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.


Solution Overview


𝛀𝟏𝟎 𝛀𝟐

𝟎

Person by Person Approach

Agent 1 Agent 2

Solution Overview


𝛀𝟏𝟎 𝛀𝟐

𝟎

𝛀𝟏𝟎


Agent 1 Agent 2

Solution Overview


𝛀𝟏𝟎 𝛀𝟐

𝟎

𝛀𝟏𝟎 𝛀𝟐

𝟏


Agent 1 Agent 2

Solution Overview


𝛀𝟏𝟎 𝛀𝟐

𝟎

𝛀𝟏𝟎 𝛀𝟐

𝟏

𝛀𝟐𝟏


Agent 1 Agent 2

Solution Overview


𝛀𝟏𝟎 𝛀𝟐

𝟎

𝛀𝟏𝟎 𝛀𝟐

𝟏

𝛀𝟏𝟏 𝛀𝟐

𝟏


Agent 1 Agent 2

Solution Overview


𝛀𝟏𝟎 𝛀𝟐

𝟎

𝛀𝟏𝟎 𝛀𝟐

𝟏

𝛀𝟏𝟏 𝛀𝟐

𝟏

𝛀𝟏∗ 𝛀𝟐

∗


Agent 1 Agent 2

Solution Overview


𝛀𝟏𝟎 𝛀𝟐

𝟎

𝛀𝟏𝟎 𝛀𝟐

𝟏

𝛀𝟏𝟏 𝛀𝟐

𝟏

𝛀𝟏∗ 𝛀𝟐

∗

Challenges & Solutions

• Constrained minimization problem


Agent 1 Agent 2

Solution Overview


𝛀𝟏𝟎 𝛀𝟐

𝟎

𝛀𝟏𝟎 𝛀𝟐

𝟏

𝛀𝟏𝟏 𝛀𝟐

𝟏

𝛀𝟏∗ 𝛀𝟐

∗




Agent 1 Agent 2

- Full centralized problem - Generalized state-partition automaton

Solution Overview


𝛀𝟏𝟎 𝛀𝟐

𝟎

𝛀𝟏𝟎 𝛀𝟐

𝟏

𝛀𝟏𝟏 𝛀𝟐

𝟏

𝛀𝟏∗ 𝛀𝟐

∗



• Converge?


Agent 1 Agent 2


Solution Overview


𝛀𝟏𝟎 𝛀𝟐

𝟎

𝛀𝟏𝟎 𝛀𝟐

𝟏

𝛀𝟏𝟏 𝛀𝟐

𝟏

𝛀𝟏∗ 𝛀𝟐

∗



• Converge?


Agent 1 Agent 2


- Yes! - Monotonicity property

Solution Overview


𝛀𝟏𝟎 𝛀𝟐

𝟎

𝛀𝟏𝟎 𝛀𝟐

𝟏

𝛀𝟏𝟏 𝛀𝟐

𝟏

𝛀𝟏∗ 𝛀𝟐

∗



• Converge?

• Minimal?


Agent 1 Agent 2



Solution Overview


𝛀𝟏𝟎 𝛀𝟐

𝟎

𝛀𝟏𝟎 𝛀𝟐

𝟏

𝛀𝟏𝟏 𝛀𝟐

𝟏

𝛀𝟏∗ 𝛀𝟐

∗



• Converge?

• Minimal?


Agent 1 Agent 2



- Yes! - Logical optimal (set inclusion)

Generalized State-Partition Automaton


• 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 ≠ ∅





∀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





∀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





∀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





∀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


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


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


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


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


• 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


Centralized Constrained Minimization Problem


• 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



• 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


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




(∃𝑞 ∈ 𝑥), 𝑞 𝑛 = 𝐾 ∧ 𝐷𝐼(ℱ𝑗(𝑞)) ≠ 𝐹-

1, otherwise


𝑄 → *0,1+ is the



-1 K

𝑃Ω𝑖 𝑠 = 𝑃Ω𝑖(𝑡)

𝑡

Agent i

𝑠

Problem Reduction




(∃𝑞 ∈ 𝑥), 𝑞 𝑛 = 𝐾 ∧ 𝐷𝐼(ℱ𝑗(𝑞)) ≠ 𝐹-

1, otherwise


𝑄 → *0,1+ is the



-1 K

𝑃Ω𝑖 𝑠 = 𝑃Ω𝑖(𝑡)

𝑡

𝓕𝒋 K Agent j

Agent i

-1

𝑠

-1 K

Problem Reduction




(∃𝑞 ∈ 𝑥), 𝑞 𝑛 = 𝐾 ∧ 𝐷𝐼(ℱ𝑗(𝑞)) ≠ 𝐹-

1, otherwise


𝑄 → *0,1+ is the



𝑠 𝑃Ω𝑖 𝑠 = 𝑃Ω𝑖

(𝑡) 𝑡

𝓕𝒋 K K-1

Agent j

Agent i

Problem Reduction


• 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


• Theorem.




• 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


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.




Synthesis Algorithm


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


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


Synthesis Algorithm


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


Synthesis Algorithm


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+ ∅


Synthesis Algorithm


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+ ∅


Synthesis Algorithm


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+ ∅


Synthesis Algorithm


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+ ∅


Correctness


• Theorem.

Let Ω ∗ be the output of Algorithm D-MIN-ACT. Then Ω ∗ is a minimal solution.

Correctness


• 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


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

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