On the Relationship between Codiagnosability and ...xiangyin/Slides/15ACCdiag.pdfXiang Yin and Stéphane Lafortune 1/17 On the Relationship between Codiagnosability and Coobservability

Xiang Yin and Stéphane Lafortune

1/17

On the Relationship between Codiagnosability and Coobservability under Dynamic Observations

EECS Department, University of Michigan

American Control Conference, July 1-3, 2015, Chicago, USA

X.Yin & S.Lafortune (UMich) July 01, 2015 ACC 2015

2/17

Introduction


Plant G

𝐷1

𝐷1(𝑠)

𝑠

𝑃1 Coordinator

𝐷𝑛(𝑠)

𝐷2(𝑠)

0

1

2

𝑃2

𝑃𝑛 𝐷2

𝐷𝑛

Decentralized Diagnosis

2/17

Introduction


Plant G

𝐷1

𝐷1(𝑠)

𝑠

𝑃1 Coordinator

𝐷𝑛(𝑠)

𝐷2(𝑠)

0

1

2

𝑃2

𝑃𝑛 𝐷2

𝐷𝑛

Plant G

𝑆1

𝑆1(𝑠)

𝑠

𝑃1 Fusion

𝑆𝑛(𝑠)

𝑆2(𝑠)

0

1

2

𝑃2

𝑃𝑛 𝑆2

𝑆𝑛

Decentralized Diagnosis Decentralized Control

2/17

Introduction


Plant G

𝐷1

𝐷1(𝑠)

𝑠

𝑃1 Coordinator

𝐷𝑛(𝑠)

𝐷2(𝑠)

0

1

2

𝑃2

𝑃𝑛 𝐷2

𝐷𝑛

Plant G

𝑆1

𝑆1(𝑠)

𝑠

𝑃1 Fusion

𝑆𝑛(𝑠)

𝑆2(𝑠)

0

1

2

𝑃2

𝑃𝑛 𝑆2

𝑆𝑛

Decentralized Diagnosis Decentralized Control

3/17

System Model


𝐺 = (𝑋, 𝐸, 𝑓, 𝑥0) is a deterministic FSA

• 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 initial state.

3/17

System Model


𝐺 = (𝑋, 𝐸, 𝑓, 𝑥0) is a deterministic FSA

• 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 initial state.

𝜔𝑖: ℒ 𝐺 → 2𝐸𝑜,𝑖

𝑃𝜔𝑖: ℒ 𝐺 → 𝐸𝑜,𝑖

∗

• A set of agents

ℐ = *1,2, … , 𝑛+

• Dynamic observations (Language-based)

• Static observations (Event-based)

∀𝑠 ∈ ℒ 𝐺 :𝜔𝑖 𝑠 = 𝐸𝑜,𝑖

𝑃𝜔𝑖 is the natural projection

4/17

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.

• A set of fault events partitioned into 𝑚 disjoint sets

𝐸𝐹 = 𝐸𝐹1 ∪ ⋯∪ 𝐸𝐹𝑚

denote by Π𝐹 the partition and ℱ = *1,… ,𝑚+ is the index set

• Ψ 𝐸𝐹𝑗 = *𝑠𝑓 ∈ ℒ 𝐺 : 𝑓 ∈ 𝐸𝐹𝑗+

4/17




• Codiagnosability:

A live language ℒ 𝐺 is said to be 𝐾-codiagnosable w.r.t. 𝜔𝑖 , 𝑖 ∈ ℐ and Π𝐹 on 𝐸𝐹 if

∀𝑗 ∈ ℱ (∀𝑠 ∈ Ψ(𝐸𝐹𝑗))(∀ 𝑡 ∈ ℒ 𝐺 /𝑠), 𝑡 ≥ 𝐾 ⇒ 𝐶𝐷-

where the codiagnosability condition 𝐶𝐷 is

∃𝑖 ∈ ℐ ∀𝜔 ∈ ℒ 𝐺 𝑃𝜔𝑖𝑠 = 𝑃𝜔𝑖

𝑡 ⇒ 𝐸𝐹𝑗 ∈ 𝜔 .

We say that ℒ 𝐺 is codiagnosable if there exists an integer 𝐾 ∈ ℕ such that it is 𝐾-codiagnosable.


• A set of fault events partitioned into 𝑚 disjoint sets

𝐸𝐹 = 𝐸𝐹1 ∪ ⋯∪ 𝐸𝐹𝑚

denote by Π𝐹 the partition and ℱ = *1,… ,𝑚+ is the index set

• Ψ 𝐸𝐹𝑗 = *𝑠𝑓 ∈ ℒ 𝐺 : 𝑓 ∈ 𝐸𝐹𝑗+

5/17



• 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. Systems, Man and Cybernetics, Part A: Systems and Humans, IEEE Transactions on, 36(2), 384-395.

• Kumar, R., & Takai, S. (2009). Inference-based ambiguity management in decentralized decision-making: Decentralized diagnosis of discrete-event systems. Automation Science and Engineering, IEEE Transactions on, 6(3), 479-491.

• Moreira, M. V., Jesus, T. C., & Basilio, J. C. (2011). Polynomial time verification of decentralized diagnosability of discrete event systems. Automatic Control, IEEE Transactions on, 56(7), 1679-1684.

• Cabasino, M. P., Giua, A., Paoli, A., & Seatzu, C. (2013). Decentralized diagnosis of discrete-event systems using labeled Petri nets. Systems, Man, and Cybernetics: Systems, IEEE Transactions on, 43(6), 1477-1485.

6/17

Decentralized Control Problem



• Specification

• Local Supervisors

ℒ 𝐻 ⊆ ℒ 𝐺

𝑆𝑖: 𝐸𝑜,𝑖∗ → Γ𝑖 , where Γ𝑖 ≔ *𝛾 ∈ 2𝐸: 𝐸 ∖ 𝐸𝑐,𝑖 ⊆ 𝛾+

• Controlled System ∧𝑖∈ℐ 𝑆𝑖/𝐺

6/17




• Coobservability :

A language ℒ 𝐻 ⊆ ℒ 𝐺 is said to be coobservable w.r.t. ℒ 𝐺 ,𝜔𝑖 and 𝐸𝑐,𝑖 , 𝑖 ∈ ℐ if

∀𝑠 ∈ ℒ 𝐻 ∀𝜎 ∈ 𝐸𝑐: 𝑠𝜎 ∈ ℒ 𝐺 ∖ ℒ 𝐻 ∃𝑖 ∈ 𝐼𝑐(𝜎)

,𝑃𝜔𝑖−1𝑃𝜔𝑖

𝑠 𝜎 ∩ ℒ 𝐻 = ∅-

where 𝐼𝑐 𝜎 ≔ *𝑖 ∈ ℐ: 𝜎 ∈ 𝐸𝑐,𝑖+.

(Conjunctive architecture)


• Specification

• Local Supervisors

ℒ 𝐻 ⊆ ℒ 𝐺

𝑆𝑖: 𝐸𝑜,𝑖∗ → Γ𝑖 , where Γ𝑖 ≔ *𝛾 ∈ 2𝐸: 𝐸 ∖ 𝐸𝑐,𝑖 ⊆ 𝛾+

• Controlled System ∧𝑖∈ℐ 𝑆𝑖/𝐺

7/17



• Rudie, K., & Wonham, W. M. (1992). Think globally, act locally: Decentralized supervisory control. Automatic Control, IEEE Transactions on, 37(11), 1692-1708.

• Yoo, T.- S., & Lafortune, S. (2002). A general architecture for decentralized supervisory control of discrete-event systems. Discrete Event Dynamic Systems, 12(3), 335-377

• Kumar, R., & Takai, S. (2007). Inference-based ambiguity management in decentralized decision-making: Decentralized control of discrete event systems. Automatic Control, IEEE Transactions on, 52(10), 1783-1794.

• Ricker, S. L., & Rudie, K. (2007). Knowledge is a terrible thing to waste: Using inference in discrete-event control problems. Automatic Control, IEEE Transactions on, 52(3), 428-441.

• Cai, K., Zhang, R., & Wonham, W. M. (2015). On relative coobservability of discrete-event systems. In American Control Conference (ACC), IEEE.

8/17

Previous Work


Control Problem

Coobservability

Diagnosis Problem

Codiagnosability

8/17

Previous Work


• Wang, W., Girard, A. R., Lafortune, S., & Lin, F. (2011). “On codiagnosability and coobservability with dynamic observations.” IEEE Transactions on Automatic Control, 56(7), 1551-1566.

Control Problem

Coobservability

Diagnosis Problem

Codiagnosability

8/17

Previous Work



Can Codiagnosability be transformed to Coobservability?

Control Problem

Coobservability

Diagnosis Problem

Codiagnosability

8/17

Previous Work



Can Codiagnosability be transformed to Coobservability?

Control Problem

Coobservability

Diagnosis Problem

Codiagnosability

• 𝐾-coodiagnosablility can be transformed to coobservablility under dynamic observations

• Coodiagnosablility can be transformed to coobservablility under static observations

9/17

Case of One Fault Type: Transformation Algorithm


0

1

3

𝑒 𝑒

𝑎

2

4

5

𝑜 𝑏

𝑎

𝑓

𝐾 = 2∀𝑠 ∈ ℒ 𝐺 :𝜔𝑖 𝑠 = *𝑜, 𝑒+

9/17



0

1

3

𝑒 𝑒

𝑎

2

4

5

𝑜 𝑏

𝑎

𝑒 𝑒

𝑓

𝑎

𝑜 𝑏

𝑎 0,-1

1,-1

2,-1

3,0

4,1

5,2

𝑯 𝟏

𝑓

𝐾 = 2∀𝑠 ∈ ℒ 𝐺 :𝜔𝑖 𝑠 = *𝑜, 𝑒+

9/17



0

1

3

𝑒 𝑒

𝑎

2

4

5

𝑜 𝑏

𝑎

𝑒 𝑒

𝑓

𝑎

𝑜 𝑏

𝑎 0,-1

1,-1

2,-1

3,0

4,1

5,2 SF

𝑯 𝟏

𝑓

𝐾 = 2∀𝑠 ∈ ℒ 𝐺 :𝜔𝑖 𝑠 = *𝑜, 𝑒+

9/17



0

1

3

𝑒 𝑒

𝑎

2

4

5

𝑜 𝑏

𝑎

𝑒 𝑒

𝑓

𝑎

𝑜 𝑏

𝑎 0,-1

1,-1

2,-1

3,0

4,1

5,2 SF 𝑐1

𝑐1

𝑐1

𝑯 𝟏

𝑓

𝐾 = 2∀𝑠 ∈ ℒ 𝐺 :𝜔𝑖 𝑠 = *𝑜, 𝑒+

9/17



0

1

3

𝑒 𝑒

𝑎

2

4

5

𝑜 𝑏

𝑎

𝑒 𝑒

𝑓

𝑎

𝑜 𝑏

𝑎 0,-1

1,-1

2,-1

3,0

4,1

5,2 SF 𝑐1

𝑐1

𝑐1

𝑯 𝟏

𝑯 𝟏

𝑓

𝐾 = 2∀𝑠 ∈ ℒ 𝐺 :𝜔𝑖 𝑠 = *𝑜, 𝑒+

9/17



0

1

3

𝑒 𝑒

𝑎

2

4

5

𝑜 𝑏

𝑎

𝑒 𝑒

𝑓

𝑎

𝑜 𝑏

𝑎 0,-1

1,-1

2,-1

3,0

4,1

5,2 USF SF 𝑐1

𝑐1

𝑐1

𝑯 𝟏

𝑯 𝟏

𝑓

𝐾 = 2∀𝑠 ∈ ℒ 𝐺 :𝜔𝑖 𝑠 = *𝑜, 𝑒+

9/17



0

1

3

𝑒 𝑒

𝑎

2

4

5

𝑜 𝑏

𝑎

𝑒 𝑒

𝑓

𝑎

𝑜 𝑏

𝑎 0,-1

1,-1

2,-1

3,0

4,1

5,2 USF SF 𝑐1 𝑐1

𝑐1

𝑐1

𝑯 𝟏

𝑯 𝟏

𝑓

𝐾 = 2∀𝑠 ∈ ℒ 𝐺 :𝜔𝑖 𝑠 = *𝑜, 𝑒+

9/17



0

1

3

𝑒 𝑒

𝑎

2

4

5

𝑜 𝑏

𝑎

𝑒 𝑒

𝑓

𝑎

𝑜 𝑏

𝑎 0,-1

1,-1

2,-1

3,0

4,1

5,2 USF SF 𝑐1 𝑐1

𝑐1

𝑐1

𝑮 𝟏 𝑯 𝟏

𝑯 𝟏

𝑓

𝐾 = 2∀𝑠 ∈ ℒ 𝐺 :𝜔𝑖 𝑠 = *𝑜, 𝑒+

10/17

Case of One Fault Type: Correctness


• Theorem. (Correctness) Language ℒ(𝐻) is 𝐾-coodiagnosable w.r.t. 𝜔𝑖 , 𝑖 ∈ ℐ and fault event set 𝐸𝐹𝑗, if

and only if, ℒ(𝐻 𝑗) is coobservable w.r.t. ℒ(𝐺 𝑗) w.r.t. 𝜔𝑖,𝐺 𝑗and 𝐸𝑐,𝑖 = *𝑐𝑗+, 𝑖 ∈ ℐ.

10/17





Sketch of the Proof: • Both can be reduced to the problem of state disambiguation.

• 𝑇𝑐𝑜𝑛𝑓 ≔ * 𝑢, 𝑣 ∈ 𝑋𝐻 𝑗 × 𝑋𝐻

𝑗: 𝑢 𝑛 = −1 ∧ 𝑣 𝑛 = 𝐾+

10/17





• Theorem. (Complexity) Let ℒ(𝐻) be the language to be diagnosed. Then the worst-case time complexity of Algorithm KCOD-COOB-I is 𝑂(𝐾|𝑋𝐻||𝐸𝐻|).

Sketch of the Proof: • Both can be reduced to the problem of state disambiguation.

• 𝑇𝑐𝑜𝑛𝑓 ≔ * 𝑢, 𝑣 ∈ 𝑋𝐻 𝑗 × 𝑋𝐻

𝑗: 𝑢 𝑛 = −1 ∧ 𝑣 𝑛 = 𝐾+

11/17

Case of Multiple Fault Types: Transformation Algorithm


12/17

Case of Multiple Fault Types: Example


1

2

3 𝑓1

𝑎, 𝑏 4

6

𝑜, 𝑏 𝑎

5

𝑏

7

8

𝑎

9

𝑓2

0

𝑜 𝑜

𝑜

𝑏

𝑎 𝐾 = 4

∀𝑠 ∈ ℒ 𝐺 :𝜔1 𝑠 = *𝑎, 𝑜+

∀𝑠 ∈ ℒ 𝐺 :𝜔2 𝑠 = *𝑏, 𝑜+

12/17



1

2

3 𝑓1

𝑎, 𝑏 4

6

𝑜, 𝑏 𝑎

5

𝑏

7

8

𝑎

9

𝑓2

0

𝑜 𝑜

𝑜

𝑏

𝑎

𝑓1

𝑎, 𝑏

𝑜, 𝑏 𝑎

𝑏

𝑎

𝑓2

𝑜 𝑜

𝑏

𝑎

0,-1

1,-1

2,-1

7,-1

9,-1

8,-1

4,1

𝑜

3,0

6,3

5,2

𝑎 6,4

𝑯 𝟏

𝐾 = 4∀𝑠 ∈ ℒ 𝐺 :𝜔1 𝑠 = *𝑎, 𝑜+

∀𝑠 ∈ ℒ 𝐺 :𝜔2 𝑠 = *𝑏, 𝑜+

12/17



1

2

3 𝑓1

𝑎, 𝑏 4

6

𝑜, 𝑏 𝑎

5

𝑏

7

8

𝑎

9

𝑓2

0

𝑜 𝑜

𝑜

𝑏

𝑎

𝑓1

𝑎, 𝑏

𝑜, 𝑏 𝑎

𝑏

𝑎

𝑓2

𝑜 𝑜

𝑏

𝑎

0,-1

1,-1

2,-1

7,-1

9,-1

8,-1

4,1

𝑜

3,0

6,3

5,2

𝑎 6,4

𝑓1

𝑎, 𝑏

𝑜, 𝑏 𝑎

𝑏

𝑎

𝑓2

𝑜 𝑜

𝑏

0,-1

1,-1

2,-1

7,0

9,2

8,1

4,-1

3,-1

6,-1

5,-1

𝑎

9,4

9,3

𝑐2

𝑜

𝑜

𝑜 𝑯 𝟏 𝑯 𝟐

𝐾 = 4∀𝑠 ∈ ℒ 𝐺 :𝜔1 𝑠 = *𝑎, 𝑜+

∀𝑠 ∈ ℒ 𝐺 :𝜔2 𝑠 = *𝑏, 𝑜+

12/17



1

2

3 𝑓1

𝑎, 𝑏 4

6

𝑜, 𝑏 𝑎

5

𝑏

7

8

𝑎

9

𝑓2

0

𝑜 𝑜

𝑜

𝑏

𝑎

𝑓1

𝑎, 𝑏

𝑜, 𝑏 𝑎

𝑏

𝑎

𝑓2

𝑜 𝑜

𝑏

𝑎

0,-1

1,-1

2,-1

7,-1

9,-1

8,-1

4,1

𝑜

3,0

6,3

5,2

𝑎 6,4

𝑓1

𝑎, 𝑏

𝑜, 𝑏 𝑎

𝑏

𝑎

𝑓2

𝑜 𝑜

𝑏

0,-1

1,-1

2,-1

7,0

9,2

8,1

4,-1

3,-1

6,-1

5,-1

𝑎

9,4

9,3

𝑐2

𝑜

𝑜

𝑜 𝑯 𝟏 𝑯 𝟐

𝐾 = 4∀𝑠 ∈ ℒ 𝐺 :𝜔1 𝑠 = *𝑎, 𝑜+

∀𝑠 ∈ ℒ 𝐺 :𝜔2 𝑠 = *𝑏, 𝑜+

12/17



1

2

3 𝑓1

𝑎, 𝑏 4

6

𝑜, 𝑏 𝑎

5

𝑏

7

8

𝑎

9

𝑓2

0

𝑜 𝑜

𝑜

𝑏

𝑎

𝑜

𝑓1

𝑎, 𝑏

𝑜, 𝑏 𝑎

𝑏

𝑎

𝑓2

𝑜 𝑜

𝑏

0,-1

0,-1

1,-1

1,-1

2,-1

2,-1

7,-1

7,0

9,-1

9,2

8,-1

8,1

4,1

4,-1

3,0

3,-1

6,3

6,-1

5,2

5,-1

𝑎

9,-1

9,4

9,-1

9,3

6,4

6,-1

𝑎

𝑜

𝑜

𝑜

𝐾 = 4∀𝑠 ∈ ℒ 𝐺 :𝜔1 𝑠 = *𝑎, 𝑜+

∀𝑠 ∈ ℒ 𝐺 :𝜔2 𝑠 = *𝑏, 𝑜+

12/17



1

2

3 𝑓1

𝑎, 𝑏 4

6

𝑜, 𝑏 𝑎

5

𝑏

7

8

𝑎

9

𝑓2

0

𝑜 𝑜

𝑜

𝑏

𝑎

𝑜

𝑓1

𝑎, 𝑏

𝑜, 𝑏 𝑎

𝑏

𝑎

𝑓2

𝑜 𝑜

𝑏

0,-1

0,-1

1,-1

1,-1

2,-1

2,-1

7,-1

7,0

9,-1

9,2

8,-1

8,1

4,1

4,-1

3,0

3,-1

6,3

6,-1

5,2

5,-1

𝑎 SF

9,-1

9,4

9,-1

9,3

6,4

6,-1

𝑎

𝑜

𝑜

𝑜

𝐾 = 4∀𝑠 ∈ ℒ 𝐺 :𝜔1 𝑠 = *𝑎, 𝑜+

∀𝑠 ∈ ℒ 𝐺 :𝜔2 𝑠 = *𝑏, 𝑜+

12/17



1

2

3 𝑓1

𝑎, 𝑏 4

6

𝑜, 𝑏 𝑎

5

𝑏

7

8

𝑎

9

𝑓2

0

𝑜 𝑜

𝑜

𝑏

𝑎

𝑜

𝑓1

𝑎, 𝑏

𝑜, 𝑏 𝑎

𝑏

𝑎

𝑓2

𝑜 𝑜

𝑏

0,-1

0,-1

1,-1

1,-1

2,-1

2,-1

7,-1

7,0

9,-1

9,2

8,-1

8,1

4,1

4,-1

3,0

3,-1

6,3

6,-1

5,2

5,-1

𝑎 SF

9,-1

9,4

9,-1

9,3

𝑐1, 𝑐2

6,4

6,-1

𝑐1, 𝑐2

𝑐1, 𝑐2

𝑎

𝑐2

𝑐2

𝑐2

𝑐2 𝑐2

𝑐1

𝑐1

𝑐1

𝑐1

𝑐1

𝑜

𝑜

𝑜

𝐾 = 4∀𝑠 ∈ ℒ 𝐺 :𝜔1 𝑠 = *𝑎, 𝑜+

∀𝑠 ∈ ℒ 𝐺 :𝜔2 𝑠 = *𝑏, 𝑜+

12/17



1

2

3 𝑓1

𝑎, 𝑏 4

6

𝑜, 𝑏 𝑎

5

𝑏

7

8

𝑎

9

𝑓2

0

𝑜 𝑜

𝑜

𝑏

𝑎

𝑜

𝑓1

𝑎, 𝑏

𝑜, 𝑏 𝑎

𝑏

𝑎

𝑓2

𝑜 𝑜

𝑏

0,-1

0,-1

1,-1

1,-1

2,-1

2,-1

7,-1

7,0

9,-1

9,2

8,-1

8,1

4,1

4,-1

3,0

3,-1

6,3

6,-1

5,2

5,-1

𝑎 SF

𝑯

9,-1

9,4

9,-1

9,3

𝑐1, 𝑐2

6,4

6,-1

𝑐1, 𝑐2

𝑐1, 𝑐2

𝑎

𝑐2

𝑐2

𝑐2

𝑐2 𝑐2

𝑐1

𝑐1

𝑐1

𝑐1

𝑐1

𝑜

𝑜

𝑜

𝐾 = 4∀𝑠 ∈ ℒ 𝐺 :𝜔1 𝑠 = *𝑎, 𝑜+

∀𝑠 ∈ ℒ 𝐺 :𝜔2 𝑠 = *𝑏, 𝑜+

12/17



1

2

3 𝑓1

𝑎, 𝑏 4

6

𝑜, 𝑏 𝑎

5

𝑏

7

8

𝑎

9

𝑓2

0

𝑜 𝑜

𝑜

𝑏

𝑎

𝑜

𝑓1

𝑎, 𝑏

𝑜, 𝑏 𝑎

𝑏

𝑎

𝑓2

𝑜 𝑜

𝑏

0,-1

0,-1

1,-1

1,-1

2,-1

2,-1

7,-1

7,0

9,-1

9,2

8,-1

8,1

4,1

4,-1

3,0

3,-1

6,3

6,-1

5,2

5,-1

𝑎 SF

USF

𝑯

9,-1

9,4

9,-1

9,3

𝑐1, 𝑐2

6,4

6,-1

𝑐1, 𝑐2

𝑐1, 𝑐2

𝑎

𝑐2

𝑐2

𝑐2

𝑐2 𝑐2

𝑐1

𝑐1

𝑐1

𝑐1

𝑐1

𝑜

𝑜

𝑜

𝐾 = 4∀𝑠 ∈ ℒ 𝐺 :𝜔1 𝑠 = *𝑎, 𝑜+

∀𝑠 ∈ ℒ 𝐺 :𝜔2 𝑠 = *𝑏, 𝑜+

12/17



1

2

3 𝑓1

𝑎, 𝑏 4

6

𝑜, 𝑏 𝑎

5

𝑏

7

8

𝑎

9

𝑓2

0

𝑜 𝑜

𝑜

𝑏

𝑎

𝑜

𝑓1

𝑎, 𝑏

𝑜, 𝑏 𝑎

𝑏

𝑎

𝑓2

𝑜 𝑜

𝑏

0,-1

0,-1

1,-1

1,-1

2,-1

2,-1

7,-1

7,0

9,-1

9,2

8,-1

8,1

4,1

4,-1

3,0

3,-1

6,3

6,-1

5,2

5,-1

𝑎 SF

USF

𝑯

9,-1

9,4

9,-1

9,3

𝑐1, 𝑐2

𝑐1

6,4

6,-1

𝑐2

𝑐1, 𝑐2

𝑐1, 𝑐2

𝑎

𝑐2

𝑐2

𝑐2

𝑐2 𝑐2

𝑐1

𝑐1

𝑐1

𝑐1

𝑐1

𝑜

𝑜

𝑜

𝐾 = 4∀𝑠 ∈ ℒ 𝐺 :𝜔1 𝑠 = *𝑎, 𝑜+

∀𝑠 ∈ ℒ 𝐺 :𝜔2 𝑠 = *𝑏, 𝑜+

12/17



1

2

3 𝑓1

𝑎, 𝑏 4

6

𝑜, 𝑏 𝑎

5

𝑏

7

8

𝑎

9

𝑓2

0

𝑜 𝑜

𝑜

𝑏

𝑎

𝑜

𝑓1

𝑎, 𝑏

𝑜, 𝑏 𝑎

𝑏

𝑎

𝑓2

𝑜 𝑜

𝑏

0,-1

0,-1

1,-1

1,-1

2,-1

2,-1

7,-1

7,0

9,-1

9,2

8,-1

8,1

4,1

4,-1

3,0

3,-1

6,3

6,-1

5,2

5,-1

𝑎 SF

USF

𝑯

𝑮 9,-1

9,4

9,-1

9,3

𝑐1, 𝑐2

𝑐1

6,4

6,-1

𝑐2

𝑐1, 𝑐2

𝑐1, 𝑐2

𝑎

𝑐2

𝑐2

𝑐2

𝑐2 𝑐2

𝑐1

𝑐1

𝑐1

𝑐1

𝑐1

𝑜

𝑜

𝑜

𝐾 = 4∀𝑠 ∈ ℒ 𝐺 :𝜔1 𝑠 = *𝑎, 𝑜+

∀𝑠 ∈ ℒ 𝐺 :𝜔2 𝑠 = *𝑏, 𝑜+

13/17

Case of Multiple Fault Types: Correctness


• Theorem. (Correctness) Language ℒ(𝐻) is 𝐾-coodiagnosable w.r.t. 𝜔𝑖 , 𝑖 ∈ ℐand Π𝐹 on 𝐸𝐹, if and only if, ℒ(𝐻 ) is coobservable w.r.t. ℒ(𝐺 ) w.r.t. 𝜔𝑖,𝐺

and 𝐸𝑐,𝑖 = *𝑐𝑗: 𝑗 ∈ ℱ+, 𝑖 ∈ ℐ.

• Theorem. (Complexity) Let ℒ(𝐻) be the language to be diagnosed. Then the worst-case time complexity of Algorithm KCOD-COOB-I is 𝑂(𝐾𝑚|𝑋𝐻||𝐸𝐻|).

14/17

Case of Event-Based Observations


• Event-Based Observation



14/17

Case of Event-Based Observations


• Theorem. Let ℒ(𝐻) be the language to be diagnosed. When the observations are event-based, codiagnosability can be transformed to coobservability in 𝑂( 𝑋𝐻|𝑚𝑛+𝑚+1 𝐸𝐻|2).

Sketch of the Proof: • ℒ(𝐻) is codiagnosable if and only if it is 𝑋𝐻 𝑛-codiagnosable • Replace the diagnosis delay 𝐾 by 𝑋𝐻 𝑛.

• Event-Based Observation



15/17

Why Study Transformation


Control Problem Diagnosis Problem

Transformation Algorithm

Solution 1 Solution 2

16/17

Application to Optimization of Sensor Activation


• Definition. (Feasibility) An observation mapping 𝜔𝑖: ℒ 𝐺 → 2𝐸𝑜,𝑖 is said to be a feasible sensor activation policy if

∀𝑠, 𝑡 ∈ ℒ 𝐺 ,𝑃𝜔𝑖𝑠 = 𝑃𝜔𝑖

𝑡 ⇒ 𝜔𝑖 𝑠 = 𝜔𝑖(𝑡)-

16/17






• Theorem. Let 𝐻 be the original system and 𝐺 be the transformed system. Then, 𝜔𝑖 is a feasible sensor activation policy for 𝐻 if and only if 𝜔𝑖,𝐺 is a feasible sensor

activation policy for 𝐺 .

16/17






• Theorem. Let 𝐻 be the original system and 𝐺 be the transformed system. Then, 𝜔𝑖 is a feasible sensor activation policy for 𝐻 if and only if 𝜔𝑖,𝐺 is a feasible sensor

activation policy for 𝐺 .

• Synthesizing an optimal sensor activation policy for 𝐾-coodiagnosablility had remained an open problem.

• Synthesizing an optimal sensor activation policy for coobservability has been solved in the literature.

• Apply our transformation algorithm.

17/17

Summary

Contributions:

• When the observation properties are language-based, 𝐾-coodiagnosablility can be transformed to coobservablility.

• When the observation properties are event-based, coodiagnosablility can be transformed to coobservablility.

• Allow the leveraging of the large existing literature on solution methodologies for problems of decentralized control to solve corresponding problems of decentralized fault diagnosis.


Wang et.al., 2011

Wang et.al., 2011

Language-Based [Co]observability

Language-Based [Co]diagnosability

Static [Co]observability

Static [Co]diagnosability

Language-Based K-[Co]diagnosability ⊇

⊇

This Work

This Work

Documents

On the Relationship between Codiagnosability and ...xiangyin/Slides/15ACCdiag.pdfXiang Yin and Stéphane Lafortune 1/17 On the Relationship between Codiagnosability and Coobservability