Minimization of Sensor Activation in Decentralized Fault Diagnosis - - PowerPoint PPT Presentation

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

𝑄2(𝑡)

𝑄2

𝑡 𝑄

1(𝑡)

𝑄

1

𝑡 𝑡

𝐸1 𝐸2

Coordinator

Fault Alarm

Plant G 1 2 3 4 5 Agent 1 Agent 2

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Introduction

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

𝑄Ω2(𝑡)

𝑄2

𝑡

Ω2

𝑄Ω1(𝑡)

𝑄

1

𝑡

Ω1

𝑡

𝐸1 𝐸2

Coordinator

Fault Alarm

Plant G 1 2 3 4 5 Agent 1 Agent 2

System Model

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𝐻 = (𝑅, Σ, 𝜀, 𝑟0) is a deterministic FSA

• 𝑅 is the finite set of states;
• Σ is the finite set of events;
• 𝜀: 𝑅 × Σ → 𝑅 is the partial transition function;
• 𝑟0 is the initial state.

System Model

• Sensor activation policy Ω = (𝐵, 𝑀), where 𝐵 = (𝑅𝐵, Σ𝑝, 𝜀𝐵, 𝑟0,𝐵) and 𝑀: 𝑅𝐵 → 2Σ𝑝;
• Projection 𝑄Ω: ℒ 𝐻 → Σ𝑝

∗

• State estimate ℰΩ

𝐻 𝑡

• Observer 𝑃𝑐𝑡Ω 𝐻 = 𝑌, Σ𝑝, 𝑔, 𝑦0 , and 𝑦 = 𝐽 𝑦 , 𝐵 𝑦

, 𝐽 𝑦 ∈ 2𝑅. 𝐵 𝑦 ∈ 𝑅𝐵

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

𝐻 = (𝑅, Σ, 𝜀, 𝑟0) is a deterministic FSA

• 𝑅 is the finite set of states;
• Σ is the finite set of events;
• 𝜀: 𝑅 × Σ → 𝑅 is the partial transition function;
• 𝑟0 is the initial state.

1 5 4 6 2 3

𝑔

7

𝑔 𝑝 𝑝 𝑝 𝑐 𝑐 𝑏 𝑏 𝑝

2 3 𝑝 𝑏 1

*𝑝+ *𝑏+ ∅

( 2,4,7 , 2)

𝑝 𝑏

( 6 , 3) ( 1,3,5,7 , 1) 𝛁 𝑷𝒄𝒕𝜵 𝑯 𝑯

Decentralized Diagnosis Problem

• A fault event 𝑓𝑒 ∈ Σ ∖ (∪𝑗=1,2 Σ𝑝,𝑗)
• Ψ 𝑓𝑒 = *𝑡𝑓𝑒 ∈ ℒ 𝐻 : 𝑡 ∈ Σ∗+

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• Two agents ℐ = *1,2+, Ω

= Ω1, Ω2 with Σ𝑝,1 and Σ𝑝,2

Decentralized Diagnosis Problem

• 𝑌 is the finite set of states;
• 𝐹 is the finite set of events;
• 𝑔: 𝑌 × 𝐹 → 𝑌 is the partial transition function;
• 𝑌0 is the set of initial states.
• K-Codiagnosability:

A live language ℒ 𝐻 is said to be 𝐿-codiagnosable w.r.t. Ω and 𝑓𝑒 if (∀𝑡 ∈ Ψ(𝑓𝑒 ))(∀𝑢 ∈ ℒ 𝐻 /𝑡), 𝑢 ≥ 𝐿 ⇒ 𝐷𝐸- where the codiagnosability condition 𝐷𝐸 is ∃𝑗 ∈ *1,2+ ∀𝜕 ∈ ℒ 𝐻 𝑄Ω𝑗 𝑥 = 𝑄Ω𝑗 𝑡𝑢 ⇒ 𝑓𝑒 ∈ 𝜕 .

• A fault event 𝑓𝑒 ∈ Σ ∖ (∪𝑗=1,2 Σ𝑝,𝑗)
• Ψ 𝑓𝑒 = *𝑡𝑓𝑒 ∈ ℒ 𝐻 : 𝑡 ∈ Σ∗+

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• Two agents ℐ = *1,2+, Ω

= Ω1, Ω2 with Σ𝑝,1 and Σ𝑝,2

Problem Formulation

• Ω

′ < Ω ∗ is defined in terms of set inclusion.

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• 𝑌 is the finite set of states;
• 𝐹 is the finite set of events;
• 𝑔: 𝑌 × 𝐹 → 𝑌 is the partial transition function;
• 𝑌0 is the set of initial states.
• Decentralized Minimization Problem

Let 𝐻 be the system with fault event 𝑓𝑒. For each agent 𝑗 ∈ 1,2 , let Σ𝑝,𝑗 ⊆ Σ be the set of observable events. Find a sensor activation policy Ω

∗ = ,Ω1 ∗, Ω2 ∗- such that

• C1. ℒ 𝐻 is 𝐿-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. 5/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

Solution Overview

6/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

𝛁𝟐

𝟏

𝛁𝟑

𝟏

Person by Person Approach

Agent 1 Agent 2

Solution Overview

6/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

𝛁𝟐

𝟏

𝛁𝟑

𝟏

𝛁𝟐

𝟏

Person by Person Approach

Agent 1 Agent 2

Solution Overview

6/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

𝛁𝟐

𝟏

𝛁𝟑

𝟏

𝛁𝟐

𝟏

𝛁𝟑

𝟐

Person by Person Approach

Agent 1 Agent 2

Solution Overview

6/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

𝛁𝟐

𝟏

𝛁𝟑

𝟏

𝛁𝟐

𝟏

𝛁𝟑

𝟐

𝛁𝟑

𝟐

Person by Person Approach

Agent 1 Agent 2

Solution Overview

6/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

𝛁𝟐

𝟏

𝛁𝟑

𝟏

𝛁𝟐

𝟏

𝛁𝟑

𝟐

𝛁𝟐

𝟐

𝛁𝟑

𝟐

Person by Person Approach

Agent 1 Agent 2

Solution Overview

6/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

𝛁𝟐

𝟏

𝛁𝟑

𝟏

𝛁𝟐

𝟏

𝛁𝟑

𝟐

𝛁𝟐

𝟐

𝛁𝟑

𝟐

Person by Person Approach

Agent 1 Agent 2

Solution Overview

6/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

𝛁𝟐

𝟏

𝛁𝟑

𝟏

𝛁𝟐

𝟏

𝛁𝟑

𝟐

𝛁𝟐

𝟐

𝛁𝟑

𝟐

𝛁𝟐

∗

𝛁𝟑

∗

Person by Person Approach

Agent 1 Agent 2

Solution Overview

6/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

𝛁𝟐

𝟏

𝛁𝟑

𝟏

𝛁𝟐

𝟏

𝛁𝟑

𝟐

𝛁𝟐

𝟐

𝛁𝟑

𝟐

𝛁𝟐

∗

𝛁𝟑

∗

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

𝛁𝟐

𝟏

𝛁𝟑

𝟏

𝛁𝟐

𝟏

𝛁𝟑

𝟐

𝛁𝟐

𝟐

𝛁𝟑

𝟐

𝛁𝟐

∗

𝛁𝟑

∗

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

𝛁𝟐

𝟏

𝛁𝟑

𝟏

𝛁𝟐

𝟏

𝛁𝟑

𝟐

𝛁𝟐

𝟐

𝛁𝟑

𝟐

𝛁𝟐

∗

𝛁𝟑

∗

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

𝛁𝟐

𝟏

𝛁𝟑

𝟏

𝛁𝟐

𝟏

𝛁𝟑

𝟐

𝛁𝟐

𝟐

𝛁𝟑

𝟐

𝛁𝟐

∗

𝛁𝟑

∗

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

𝛁𝟐

𝟏

𝛁𝟑

𝟏

𝛁𝟐

𝟏

𝛁𝟑

𝟐

𝛁𝟐

𝟐

𝛁𝟑

𝟐

𝛁𝟐

∗

𝛁𝟑

∗

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

𝛁𝟐

𝟏

𝛁𝟑

𝟏

𝛁𝟐

𝟏

𝛁𝟑

𝟐

𝛁𝟐

𝟐

𝛁𝟑

𝟐

𝛁𝟐

∗

𝛁𝟑

∗

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

7/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

• Generalized State-Partition Automaton

Let 𝐻 be an automaton, Ω a sensor activation policy and 𝑃𝑐𝑡Ω 𝐻 be the corresponding observer. We sat that 𝐻 is a state-partition automaton (SPA) w.r.t. Ω, if ∀𝑦, 𝑧 ∈ 𝑌: 𝐽 𝑦 = 𝐽 𝑧 or 𝐽 𝑦 ∩ 𝐽 𝑧 ≠ ∅

Generalized State-Partition Automaton

7/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

• Generalized State-Partition Automaton

Let 𝐻 be an automaton, Ω a sensor activation policy and 𝑃𝑐𝑡Ω 𝐻 be the corresponding observer. We sat that 𝐻 is a state-partition automaton (SPA) w.r.t. Ω, if ∀𝑦, 𝑧 ∈ 𝑌: 𝐽 𝑦 = 𝐽 𝑧 or 𝐽 𝑦 ∩ 𝐽 𝑧 ≠ ∅

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

7/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

• Generalized State-Partition Automaton

Let 𝐻 be an automaton, Ω a sensor activation policy and 𝑃𝑐𝑡Ω 𝐻 be the corresponding observer. We sat that 𝐻 is a state-partition automaton (SPA) w.r.t. Ω, if ∀𝑦, 𝑧 ∈ 𝑌: 𝐽 𝑦 = 𝐽 𝑧 or 𝐽 𝑦 ∩ 𝐽 𝑧 ≠ ∅

( 7 , 1)

𝑐 𝑐

( 5 , 1) ( 1,2,3,4,6 , 1) 1 5 4 6 2 3

𝑔

7

𝑔 𝑝 𝑝 𝑐 𝑐 𝑏 𝑏 𝑝

1 𝑐

*𝑐+

Ω1

Generalized State-Partition Automaton

7/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

• Generalized State-Partition Automaton

Let 𝐻 be an automaton, Ω a sensor activation policy and 𝑃𝑐𝑡Ω 𝐻 be the corresponding observer. We sat that 𝐻 is a state-partition automaton (SPA) w.r.t. Ω, if ∀𝑦, 𝑧 ∈ 𝑌: 𝐽 𝑦 = 𝐽 𝑧 or 𝐽 𝑦 ∩ 𝐽 𝑧 ≠ ∅

( 7 , 1)

𝑐 𝑐

( 5 , 1) ( 1,2,3,4,6 , 1) 1 5 4 6 2 3

𝑔

7

𝑔 𝑝 𝑝 𝑝 𝑐 𝑐 𝑏 𝑏 𝑝

1 𝑐

*𝑐+

2 3 𝑝 𝑏 1

*𝑝+ *𝑏+ ∅

( 2,4, 𝟖 , 2)

𝑝 𝑏

( 6 , 3) ( 1,3,5, 𝟖 , 1)

Ω1 Ω2

Generalized State-Partition Automaton

7/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

• Generalized State-Partition Automaton

Let 𝐻 be an automaton, Ω a sensor activation policy and 𝑃𝑐𝑡Ω 𝐻 be the corresponding observer. We sat that 𝐻 is a state-partition automaton (SPA) w.r.t. Ω, if ∀𝑦, 𝑧 ∈ 𝑌: 𝐽 𝑦 = 𝐽 𝑧 or 𝐽 𝑦 ∩ 𝐽 𝑧 ≠ ∅

( 7 , 1)

𝑐 𝑐

( 5 , 1) ( 1,2,3,4,6 , 1) 1 5 4 6 2 3

𝑔

7

𝑔 𝑝 𝑝 𝑝 𝑐 𝑐 𝑏 𝑏 𝑝

1 𝑐

*𝑐+

2 3 𝑝 𝑏 1

*𝑝+ *𝑏+ ∅

( 2,4,7 , 2)

𝑝 𝑏

( 6 , 3) ( 1,3,5,7 , 1)

• Theorem

Let 𝐻 be the system automaton, Ω be a sensor activation policy. Then 𝑃𝑐𝑡Ω

+ 𝐻 ∥ 𝐻 is an SPA w.r.t. Ω such that ℒ 𝑃𝑐𝑡Ω + 𝐻 ∥ 𝐻 = ℒ 𝐻 .

Ω1 Ω2

Inference Function

8/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

Suppose that 𝐻 is an SPA w.r.t. 𝛻 and 𝑃𝑐𝑡𝛻 𝐻 = (𝑌, Σ𝑝, 𝑔, 𝑦0) is the

• bserver. Then for any state q ∈ 𝑅, there exists a unique information

state ℱ 𝑟 ∈ 2𝑅 s.t. 𝑟 ∈ ℱ 𝑟 and ∃𝑟𝐵 ∈ 𝑅𝐵: ℱ 𝑟 , 𝑟𝐵 ∈ 𝑌 We call this information state ℱ 𝑟 the inference of state 𝑟. ℱ: 𝑅 → 2𝑅 such that ∀𝑡 ∈ ℒ 𝐻 : 𝜀 𝑟0, 𝑡 = 𝑟 ⇒ ,ℱ 𝑟 = 𝐽(𝑔(𝑄Ω(𝑡)))-

( 7 , 1)

𝑐 𝑐

( 5 , 1) ( 1,2,3,4,6 , 1) 1 5 4 6 2 3

𝑔

7

𝑔 𝑝 𝑝 𝑝 𝑐 𝑐 𝑏 𝑏 𝑝

1 𝑐

*𝑐+

• Inference Function

Inference Function

8/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

Suppose that 𝐻 is an SPA w.r.t. 𝛻 and 𝑃𝑐𝑡𝛻 𝐻 = (𝑌, Σ𝑝, 𝑔, 𝑦0) is the

• bserver. Then for any state q ∈ 𝑅, there exists a unique information

state ℱ 𝑟 ∈ 2𝑅 s.t. 𝑟 ∈ ℱ 𝑟 and ∃𝑟𝐵 ∈ 𝑅𝐵: ℱ 𝑟 , 𝑟𝐵 ∈ 𝑌 We call this information state ℱ 𝑟 the inference of state 𝑟. ℱ: 𝑅 → 2𝑅 such that ∀𝑡 ∈ ℒ 𝐻 : 𝜀 𝑟0, 𝑡 = 𝑟 ⇒ ,ℱ 𝑟 = 𝐽(𝑔(𝑄Ω(𝑡)))-

𝑐 𝑐

1 5 4 6 2 3

𝑔

7

𝑔 𝑝 𝑝 𝑝 𝑐 𝑐 𝑏 𝑏 𝑝

1 𝑐

*𝑐+

• Inference Function

( 1,2,3,4,6 , 1) ( 5 , 1) ( 7 , 1)

Problem Reformulation

9/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

𝐻 = (𝑅 , Σ, 𝜀 , 𝑟 0) is a deterministic FSA

• 𝑅

= 𝑅 × −1,0,1, … , 𝐿 and 𝑟 0 = 𝑟0, −1 . K-Augmented Automaton

1 5 4 6 2 3

𝑔

7

𝑔 𝑝 𝑝 𝑝 𝑐 𝑐 𝑏 𝑏 𝑝 𝑔 𝑔 𝑝 𝑝 𝑝 𝑐 𝑐 𝑏 𝑏 𝑝 2

,

• 1

1 ,

• 1

4 , 6 , 1 3 , 5 , 1 7 , 1

𝑯 𝑯 𝑳 = 𝟐

Problem Reformulation

9/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

𝐻 = (𝑅 , Σ, 𝜀 , 𝑟 0) is a deterministic FSA

• 𝑅

= 𝑅 × −1,0,1, … , 𝐿 and 𝑟 0 = 𝑟0, −1 . K-Augmented Automaton

• 𝐸𝐽 𝑦 = 𝑂 if ∀𝑟 ∈ 𝑦: 𝑟 𝑜 = −1
• 𝐸𝐽 𝑦 = 𝐺 if ∀𝑟 ∈ 𝑦: 𝑟 𝑜 ≥ 0
• 𝐸𝐽 𝑦 = 𝐷1 if ,∀𝑟 ∈ 𝑦: 𝑟 𝑜 ≠ 𝐿- ∧ ,∃𝑟, 𝑟′ ∈ 𝑦: 𝑟 𝑜 = −1 ∧ 0 ≤ 𝑟′ 𝑜 < 𝐿-
• 𝐸𝐽 𝑦 = 𝐷2 if ∃𝑟, 𝑟′ ∈ 𝑦: 𝑟 𝑜 = −1 ∧ 𝑟′ 𝑜 = 𝐿

Diagnosability Function

Centralized Constrained Minimization Problem

• 𝑌 is the finite set of states;
• 𝐹 is the finite set of events;
• 𝑔: 𝑌 × 𝐹 → 𝑌 is the partial transition function;
• 𝑌0 is the set of initial states.
• Centralized Constrained Minimization Problem

Let 𝑗, 𝑘 ∈ 1,2 , 𝑗 ≠ 𝑘 be two agent. Suppose that the sensor activation policy Ω𝑘

for Agent 𝑘 is fixed. Find a sensor activation policy Ω𝑗 for Agent 𝑗

s.t.

• C1. ℒ 𝐻 is 𝐿-codiagnosable w.r.t. Ω1

, Ω2 ;

• C2. For any Ω𝑗

′ satisfying (C1), we have Ω𝑗 ′ ≮ Ω𝑗

10/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

Centralized Constrained Minimization Problem

• 𝑌 is the finite set of states;
• 𝐹 is the finite set of events;
• 𝑔: 𝑌 × 𝐹 → 𝑌 is the partial transition function;
• 𝑌0 is the set of initial states.
• Centralized Constrained Minimization Problem

Let 𝑗, 𝑘 ∈ 1,2 , 𝑗 ≠ 𝑘 be two agent. Suppose that the sensor activation policy Ω𝑘

for Agent 𝑘 is fixed. Find a sensor activation policy Ω𝑗 for Agent 𝑗

s.t.

• C1. ℒ 𝐻 is 𝐿-codiagnosable w.r.t. Ω1

, Ω2 ;

• C2. For any Ω𝑗

′ satisfying (C1), we have Ω𝑗 ′ ≮ Ω𝑗

10/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

• 𝑌 is the finite set of states;
• 𝐹 is the finite set of events;
• 𝑔: 𝑌 × 𝐹 → 𝑌 is the partial transition function;
• 𝑌0 is the set of initial states.
• Centralized Sensor Minimization Problem for IS-Based Property

Let 𝐻 = (𝑅, Σ, 𝜀, 𝑟0) be the system and 𝜚: 2𝑅 → *0,1+ be a function on information states. Find a sensor activation policy Ω s.t.

• C1. ∀𝑡 ∈ ℒ 𝐻 : 𝜚 ℰΩ

𝐻 𝑡

= 1;

• C2. For any Ω′ satisfying (C1), we have Ω′ ≮ Ω .

Problem Reduction

11/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

𝐷𝐸𝑗 𝑦 = 0, if 𝐸𝐽 𝑦 = 𝐷2 𝑏𝑜𝑒 (∃𝑟 ∈ 𝑦), 𝑟 𝑜 = 𝐿 ∧ 𝐸𝐽(ℱ

𝑘(𝑟)) ≠ 𝐺-

1,

• therwise

Suppose that 𝐻 = 𝑅 , Σ, 𝜀 , 𝑟 0 is a SPA w.r.t. Ω𝑘

and ℱ 𝑘: 2𝑅 → *0,1+ is the

corresponding inference function. We define the codiagnosability function 𝐷𝐸𝑗 𝑦 : 2𝑅

→ *0,1+ for Agent i as follows. For each 𝑦 ∈ 2𝑅

Problem Reduction

11/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

𝐷𝐸𝑗 𝑦 = 0, if 𝐸𝐽 𝑦 = 𝐷2 𝑏𝑜𝑒 (∃𝑟 ∈ 𝑦), 𝑟 𝑜 = 𝐿 ∧ 𝐸𝐽(ℱ

𝑘(𝑟)) ≠ 𝐺-

1,

• therwise

Suppose that 𝐻 = 𝑅 , Σ, 𝜀 , 𝑟 0 is a SPA w.r.t. Ω𝑘

and ℱ 𝑘: 2𝑅 → *0,1+ is the

corresponding inference function. We define the codiagnosability function 𝐷𝐸𝑗 𝑦 : 2𝑅

→ *0,1+ for Agent i as follows. For each 𝑦 ∈ 2𝑅

• 1

K

𝑄Ω𝑗 𝑡 = 𝑄Ω𝑗(𝑢) 𝑢

Agent i

𝑡

Problem Reduction

11/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

𝐷𝐸𝑗 𝑦 = 0, if 𝐸𝐽 𝑦 = 𝐷2 𝑏𝑜𝑒 (∃𝑟 ∈ 𝑦), 𝑟 𝑜 = 𝐿 ∧ 𝐸𝐽(ℱ

𝑘(𝑟)) ≠ 𝐺-

1,

• therwise

Suppose that 𝐻 = 𝑅 , Σ, 𝜀 , 𝑟 0 is a SPA w.r.t. Ω𝑘

and ℱ 𝑘: 2𝑅 → *0,1+ is the

corresponding inference function. We define the codiagnosability function 𝐷𝐸𝑗 𝑦 : 2𝑅

→ *0,1+ for Agent i as follows. For each 𝑦 ∈ 2𝑅

• 1

K

𝑄Ω𝑗 𝑡 = 𝑄Ω𝑗(𝑢) 𝑢

𝓖𝒌 K

Agent j Agent i

• 1

𝑡

Problem Reduction

11/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

𝐷𝐸𝑗 𝑦 = 0, if 𝐸𝐽 𝑦 = 𝐷2 𝑏𝑜𝑒 (∃𝑟 ∈ 𝑦), 𝑟 𝑜 = 𝐿 ∧ 𝐸𝐽(ℱ

𝑘(𝑟)) ≠ 𝐺-

1,

• therwise

Suppose that 𝐻 = 𝑅 , Σ, 𝜀 , 𝑟 0 is a SPA w.r.t. Ω𝑘

and ℱ 𝑘: 2𝑅 → *0,1+ is the

corresponding inference function. We define the codiagnosability function 𝐷𝐸𝑗 𝑦 : 2𝑅

→ *0,1+ for Agent i as follows. For each 𝑦 ∈ 2𝑅

• 1

K

𝑄Ω𝑗 𝑡 = 𝑄Ω𝑗(𝑢) 𝑢

𝓖𝒌 K

Agent j Agent i

• 1

𝑡

• 1

K

Problem Reduction

11/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

𝐷𝐸𝑗 𝑦 = 0, if 𝐸𝐽 𝑦 = 𝐷2 𝑏𝑜𝑒 (∃𝑟 ∈ 𝑦), 𝑟 𝑜 = 𝐿 ∧ 𝐸𝐽(ℱ

𝑘(𝑟)) ≠ 𝐺-

1,

• therwise

Suppose that 𝐻 = 𝑅 , Σ, 𝜀 , 𝑟 0 is a SPA w.r.t. Ω𝑘

and ℱ 𝑘: 2𝑅 → *0,1+ is the

corresponding inference function. We define the codiagnosability function 𝐷𝐸𝑗 𝑦 : 2𝑅

→ *0,1+ for Agent i as follows. For each 𝑦 ∈ 2𝑅

𝑡 𝑄Ω𝑗 𝑡 = 𝑄Ω𝑗(𝑢) 𝑢

𝓖𝒌 K

K-1

Agent j Agent i

Problem Reduction

12/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

• Theorem.

Suppose that 𝐻 = 𝑅 , Σ, 𝜀 , 𝑟 0 is a SPA w.r.t. Ω𝑘

. Then ℒ 𝐻 is 𝐿-codiagnosable

w.r.t. Ω1

, Ω2 and 𝑓𝑒, if and only if,

∀𝑡 ∈ ℒ 𝐻 : 𝐷𝐸𝑗 ℰΩ

𝐻 𝑡

= 1

Problem Reduction

12/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

• Theorem.

Suppose that 𝐻 = 𝑅 , Σ, 𝜀 , 𝑟 0 is a SPA w.r.t. Ω𝑘

. Then ℒ 𝐻 is 𝐿-codiagnosable

w.r.t. Ω1

, Ω2 and 𝑓𝑒, if and only if,

∀𝑡 ∈ ℒ 𝐻 : 𝐷𝐸𝑗 ℰΩ

𝐻 𝑡

= 1

• Centralized Sensor Minimization Problem for IS-Based Property

Let 𝐻 = (𝑅, Σ, 𝜀, 𝑟0) be the system and 𝜚: 2𝑅 → *0,1+ be a function on information states. Find a sensor activation policy Ω s.t.

• C1. ∀𝑡 ∈ ℒ 𝐻 : 𝜚(ℰΩ

𝐻 𝑡 ) = 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 𝐻 = 𝑅 , Σ, 𝜀 , 𝑟 0 is a SPA w.r.t. Ω𝑘

. Then ℒ 𝐻 is 𝐿-codiagnosable

w.r.t. Ω1

, Ω2 and 𝑓𝑒, if and only if,

∀𝑡 ∈ ℒ 𝐻 : 𝐷𝐸𝑗 ℰΩ

𝐻 𝑡

= 1

Synthesis Algorithm

13/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

𝑔 𝑔 𝑝 𝑝 𝑐 𝑐 𝑏 𝑏 𝑝 2

,

• 1

1 ,

• 1

4 , 6 , 1 3 , 5 , 1 7 , 1

1 𝑐

*𝑐+

1 𝑏, 𝑝

*𝑏, 𝑝+ 𝑝 Agent 1:𝚻𝒑,𝟐 = *𝒄+ Agent 2:𝚻𝒑,𝟐 = *𝒑, 𝒃+

Synthesis Algorithm

13/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

𝑔 𝑔 𝑝 𝑝 𝑐 𝑐 𝑏 𝑏 𝑝 2

,

• 1

1 ,

• 1

4 , 6 , 1 3 , 5 , 1 7 , 1

1 𝑐

*𝑐+

1 𝑏, 𝑝

*𝑏, 𝑝+

1 𝑐

*𝑐+ 𝑝 Agent 1:𝚻𝒑,𝟐 = *𝒄+ Agent 2:𝚻𝒑,𝟐 = *𝒑, 𝒃+

Synthesis Algorithm

13/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

𝑔 𝑔 𝑝 𝑝 𝑐 𝑐 𝑏 𝑏 𝑝 2

,

• 1

1 ,

• 1

4 , 6 , 1 3 , 5 , 1 7 , 1

1 𝑐

*𝑐+

2 3 𝑝 𝑏 1

*𝑝+ *𝑏+ ∅

1 𝑏, 𝑝

*𝑏, 𝑝+

1 𝑐

*𝑐+ 𝑝 Agent 1:𝚻𝒑,𝟐 = *𝒄+ Agent 2:𝚻𝒑,𝟐 = *𝒑, 𝒃+

Synthesis Algorithm

13/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

𝑔 𝑔 𝑝 𝑝 𝑐 𝑐 𝑏 𝑏 𝑝 2

,

• 1

1 ,

• 1

4 , 6 , 1 3 , 5 , 1 7 , 1

1 𝑐

*𝑐+

2 3 𝑝 𝑏 1

*𝑝+ *𝑏+ ∅

1 𝑏, 𝑝

*𝑏, 𝑝+

1 𝑐

*𝑐+ 𝑝

2 3 𝑝 𝑏 1

*𝑝+ *𝑏+ ∅ Agent 1:𝚻𝒑,𝟐 = *𝒄+ Agent 2:𝚻𝒑,𝟐 = *𝒑, 𝒃+

Synthesis Algorithm

13/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

𝑔 𝑔 𝑝 𝑝 𝑐 𝑐 𝑏 𝑏 𝑝 2

,

• 1

1 ,

• 1

4 , 6 , 1 3 , 5 , 1 7 , 1

1 𝑐

*𝑐+

2 3 𝑝 𝑏 1

*𝑝+ *𝑏+ ∅

( 2,4, 𝟖 , 2)

𝑝 𝑏

( 6 , 3) ( 1,3,5, 𝟖 , 1)

1 𝑏, 𝑝

*𝑏, 𝑝+

1 𝑐

*𝑐+ 𝑝

2 3 𝑝 𝑏 1

*𝑝+ *𝑏+ ∅ Agent 1:𝚻𝒑,𝟐 = *𝒄+ Agent 2:𝚻𝒑,𝟐 = *𝒑, 𝒃+

Synthesis Algorithm

13/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

𝑔 𝑔 𝑝 𝑝 𝑐 𝑐 𝑏 𝑏 𝑝 2

,

• 1

1 ,

• 1

4 , 6 , 1 3 , 5 , 1 7 , 1

1 𝑐

*𝑐+

2 3 𝑝 𝑏 1

*𝑝+ *𝑏+ ∅

( 2,4,7′ , 2)

𝑝 𝑏

( 6 , 3) ( 1,3,5,7 , 1)

1 𝑏, 𝑝

*𝑏, 𝑝+

1 𝑐

*𝑐+ 𝑔 𝑔 𝑝 𝑝 𝑐 𝑐 𝑏 𝑏 𝑝 2

,

• 1

1 ,

• 1

4 , 6 , 1 3 , 5 , 1 7 , 1

𝑝 𝑝 7’

, 1

𝑝

2 3 𝑝 𝑏 1

*𝑝+ *𝑏+ ∅ Agent 1:𝚻𝒑,𝟐 = *𝒄+ Agent 2:𝚻𝒑,𝟐 = *𝒑, 𝒃+

Synthesis Algorithm

13/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

𝑔 𝑔 𝑝 𝑝 𝑐 𝑐 𝑏 𝑏 𝑝 2

,

• 1

1 ,

• 1

4 , 6 , 1 3 , 5 , 1 7 , 1

1 𝑐

*𝑐+

2 3 𝑝 𝑏 1

*𝑝+ *𝑏+ ∅

1 𝑏, 𝑝

*𝑏, 𝑝+

1 𝑐

*𝑐+ 𝑝

2 3 𝑝 𝑏 1

*𝑝+ *𝑏+ ∅

2 𝑐 1

*𝑐+ ∅ Agent 1:𝚻𝒑,𝟐 = *𝒄+ Agent 2:𝚻𝒑,𝟐 = *𝒑, 𝒃+

Synthesis Algorithm

13/15 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

𝑔 𝑔 𝑝 𝑝 𝑐 𝑐 𝑏 𝑏 𝑝 2

,

• 1

1 ,

• 1

4 , 6 , 1 3 , 5 , 1 7 , 1

1 𝑐

*𝑐+

2 3 𝑝 𝑏 1

*𝑝+ *𝑏+ ∅

1 𝑏, 𝑝

*𝑏, 𝑝+

1 𝑐

*𝑐+ 𝑝

2 3 𝑝 𝑏 1

*𝑝+ *𝑏+ ∅

2 𝑐 1

*𝑐+ ∅ Agent 1:𝚻𝒑,𝟐 = *𝒄+ Agent 2:𝚻𝒑,𝟐 = *𝒑, 𝒃+

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