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A General Approach for Solving Dynamic Sensor Activation Problems for a Class of Properties Xiang Yin and Stphane Lafortune EECS Department, University of Michigan 54th IEEE CDC, Dec 15-18, 2015, Osaka, Japan 0/17 X.Yin & S.Lafortune

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Xiang Yin and StΓ©phane Lafortune

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A General Approach for Solving Dynamic Sensor Activation Problems for a Class of Properties

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

Plant G 1 2 3 4 5

𝑄

  • Dynamic Sensor Activation Problem

Observer

𝑄(𝑑) 𝑑

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Introduction

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

Plant G 1 2 3 4 5 𝑑

𝑄

  • Dynamic Sensor Activation Problem

Observer

𝑄

πœ•(𝑑)

𝝏

Sensor Activation Module

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Introduction

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

Plant G 1 2 3 4 5 𝑑

𝑄

  • Dynamic Sensor Activation Problem

Observer

𝑄

πœ•(𝑑)

𝝏

Property √

Sensor Activation Module

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System Model

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

  • 𝑅 is the finite set of states;
  • Ξ£ is the finite set of events;
  • πœ€: 𝑅 Γ— Ξ£ β†’ 𝑅 is the partial transition function;
  • π‘Ÿ0 is the initial state

𝐻 = (𝑅, Ξ£, πœ€, π‘Ÿ0) is a deterministic FSA

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System Model

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

  • 𝑅 is the finite set of states;
  • Ξ£ is the finite set of events;
  • πœ€: 𝑅 Γ— Ξ£ β†’ 𝑅 is the partial transition function;
  • π‘Ÿ0 is the initial state

𝐻 = (𝑅, Ξ£, πœ€, π‘Ÿ0) is a deterministic FSA

  • Ξ£ = Σ𝑝 βˆͺ

Σ𝑑 βˆͺ Σ𝑣𝑝

  • A sensing decision is a set of events πœ„ ∈ 2Ξ£ s.t. Σ𝑝 βŠ† πœ„ βŠ† Σ𝑝 βˆͺ Σ𝑑

Θ denotes the set of sensing decisions

  • Information mapping πœ•: β„’ 𝐻 β†’ Θ

𝑄

πœ•: β„’ 𝐻 β†’ Σ𝑝 βˆͺ Σ𝑑 βˆ— denotes the corresponding projection

  • A sensor activation policy is an information mapping πœ•: β„’ 𝐻 β†’ Θ s.t.

βˆ€π‘‘, 𝑒 ∈ β„’ 𝐻 : 𝑄

πœ• 𝑑 = 𝑄 πœ• 𝑒 β‡’ πœ• 𝑑 = πœ•(𝑒)

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Information-State-Based Property

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

  • Information-State-Based Property

Let 𝐻 be the system automaton and πœ•: β„’ 𝐻 β†’ Θ be a sensor activation policy. An IS-based property w.r.t. 𝐻 is a function πœ’: 2𝑅 β†’ *0,1+ We say that πœ• satisfies πœ’ w.r.t. 𝐻, denoted by πœ• ⊨𝐻 πœ’, if βˆ€π‘‘ ∈ β„’ 𝐻 : πœ’(β„°πœ•

𝐻 𝑑 ) = 1

where β„°πœ•

𝐻 𝑑 = *π‘Ÿ ∈ 𝑅: βˆƒπ‘’ ∈ 𝑀 𝑑. 𝑒. 𝑄 πœ• 𝑒 = 𝑄 πœ• 𝑑 ∧ πœ€ π‘Ÿ0, 𝑒 = π‘Ÿ+

Information State: a set of states, 𝐽 ≔ 2𝑅

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IS-based Properties

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

  • Information-State-Based Properties
  • Opacity: privacy applications
  • Diagnosability: fault detection and isolation
  • Predictability: fault prognosis
  • Detectability: state estimation
  • Anonymity: privacy applications
  • Etc.
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Example 1

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

1 6 4 7 5

𝑝

2

𝜏1

3

𝑝 𝑔 𝑓 𝑝 𝜏2 𝑓, 𝜏1 𝑓

  • IS-based Property πœ’: 2𝑅 β†’ *0,1+

βˆ€ 𝑗 ∈ 2𝑅: πœ’ 𝑗 = 1 ⇔ ,βˆ„π‘Ÿ ∈ 1,4,5,6 : 3, π‘Ÿ βŠ† 𝑗-

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Example 1

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

1 6 4 7 5

𝑝

2

𝜏1

3

𝑝 𝑔 𝑓 𝑝 𝜏2 𝑓, 𝜏1 𝑓

  • IS-based Property πœ’: 2𝑅 β†’ *0,1+

βˆ€ 𝑗 ∈ 2𝑅: πœ’ 𝑗 = 1 ⇔ ,βˆ„π‘Ÿ ∈ 1,4,5,6 : 3, π‘Ÿ βŠ† 𝑗-

  • Sensor Activation Policy πœ•: β„’ 𝐻 β†’ Θ

βˆ€π‘‘ ∈ β„’ 𝐻 : πœ• 𝑑 = *𝑝+

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Example 1

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

1 6 4 7 5

𝑝

2

𝜏1

3

𝑝 𝑔 𝑓 𝑝 𝜏2 𝑓, 𝜏1 𝑓

  • IS-based Property πœ’: 2𝑅 β†’ *0,1+

βˆ€ 𝑗 ∈ 2𝑅: πœ’ 𝑗 = 1 ⇔ ,βˆ„π‘Ÿ ∈ 1,4,5,6 : 3, π‘Ÿ βŠ† 𝑗-

  • Sensor Activation Policy πœ•: β„’ 𝐻 β†’ Θ

βˆ€π‘‘ ∈ β„’ 𝐻 : πœ• 𝑑 = *𝑝+

  • The IS-based property is not satisfied, i.e., πœ• ⊭ πœ’

β„°πœ•

𝐻 𝑓𝑝 = *πŸ’, πŸ•+

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Example 1

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

  • IS-based Property πœ’: 2𝑅 β†’ *0,1+

βˆ€ 𝑗 ∈ 2𝑅: πœ’ 𝑗 = 1 ⇔ ,βˆ„π‘Ÿ ∈ 1,4,5,6 : 3, π‘Ÿ βŠ† 𝑗-

  • Sensor Activation Policy πœ•: β„’ 𝐻 β†’ Θ

βˆ€π‘‘ ∈ β„’ 𝐻 : πœ• 𝑑 = *𝑝+

  • The IS-based property is not satisfied, i.e., πœ• ⊭ πœ’

β„°πœ•

𝐻 𝑓𝑝 = *πŸ’, πŸ•+

  • State-disambiguation problem

1 6 4 7 5

𝑝

2

𝜏1

3

𝑝 𝑔 𝑓 𝑝 𝜏2 𝑓, 𝜏1 𝑓

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Example 2

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

S NS 𝑑 𝑒 𝑄

πœ• 𝑑 = 𝑄 πœ•(𝑒)

  • IS-based Property πœ’: 2𝑅 β†’ *0,1+

βˆ€ 𝑗 ∈ 2𝑅: πœ’ 𝑗 = 1 ⇔ ,𝑗 ⊈ π‘Œπ‘‡π‘“π‘‘π‘ π‘“π‘’-, where π‘Œπ‘‡π‘“π‘‘π‘ π‘“π‘’ βŠ† π‘Œ

  • Opacity problem
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Problem Formulation

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

  • Minimal Sensor Activation Problem for IS-Based Properties

Let 𝐻 = (𝑅, Ξ£, πœ€, π‘Ÿ0) be the system automaton and πœ’: 2𝑅 β†’ *0,1+ be an IS-based property w.r.t. 𝐻. Find a sensor activation policy πœ• such that (i) πœ• ⊨𝐻 πœ’ (IS-based Property) (ii) βˆ„πœ•β€² ∈ Ξ© such that πœ• ⊨𝐻 πœ’ and πœ•β€² < πœ•. (Minimality) The Maximal Sensor Activation Problem is also defined analogously. πœ•β€² < πœ• is defined in terms of set inclusion.

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Literature Review

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.

  • 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.
  • 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.

  • Dallal, E., & Lafortune, S. (2014). On most permissive observers in dynamic sensor activation problems. Automatic

Control, IEEE Transactions on, 59(4), 966-981.

  • Sears, D., & Rudie, K. (2015). Minimal sensor activation and minimal communication in discrete-event systems.

Discrete Event Dynamic Systems, 1-55. 8/17 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

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Literature Review

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.

  • 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.
  • 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.

  • Dallal, E., & Lafortune, S. (2014). On most permissive observers in dynamic sensor activation problems. Automatic

Control, IEEE Transactions on, 59(4), 966-981.

  • Sears, D., & Rudie, K. (2015). Minimal sensor activation and minimal communication in discrete-event systems.

Discrete Event Dynamic Systems, 1-55. 8/17 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

  • Different approaches for different properties
  • The sensor activation problems for some properties have not been considered
  • Need general approach
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Bipartite Dynamic Observer

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

A bipartite dynamic observer 𝒫 w.r.t. G is a 7-tuple 𝒫 = (𝑅𝑍

𝒫, π‘…π‘Ž 𝒫, β„Žπ‘π‘Ž 𝒫 , β„Žπ‘Žπ‘ 𝒫 , 𝐹, Ξ“, 𝑧0 )

  • Bipartite Dynamic Observer (BDO)
  • 𝑅𝑍

𝒫 βŠ† 𝐽 is the set of Y-states;

  • π‘…π‘Ž

𝒫 βŠ† 𝐽 Γ— Θ is the set of Z-states so that z = (𝐽 𝑨 , Θ 𝑨 );

  • β„Žπ‘π‘Ž

𝒫 : 𝑅𝑍 𝒫 Γ— Θ β†’ Qπ‘Ž 𝒫 represents the unobservable reach;

  • β„Žπ‘Žπ‘

𝒫 : π‘…π‘Ž 𝒫 Γ— Ξ£ β†’ Q𝑍 𝒫 represents the observable reach;

  • 𝑧0

= *π‘Ÿ0+ is the initial state.

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Bipartite Dynamic Observer

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

Σ𝑝 = 𝑝 , Σ𝑑 = 𝜏1, 𝜏2 , Σ𝑣𝑝 = *𝑓, 𝑔+ 1 6 4 7 5 𝑝 2

𝜏1

3

𝑝 𝑔 𝑓 𝑝 𝜏2 𝑓, 𝜏1 𝑓

*1+ A bipartite dynamic observer 𝒫 w.r.t. G is a 7-tuple 𝒫 = (𝑅𝑍

𝒫, π‘…π‘Ž 𝒫, β„Žπ‘π‘Ž 𝒫 , β„Žπ‘Žπ‘ 𝒫 , 𝐹, Ξ“, 𝑧0 )

  • Bipartite Dynamic Observer (BDO)
  • 𝑅𝑍

𝒫 βŠ† 𝐽 is the set of Y-states;

  • π‘…π‘Ž

𝒫 βŠ† 𝐽 Γ— Θ is the set of Z-states so that z = (𝐽 𝑨 , Θ 𝑨 );

  • β„Žπ‘π‘Ž

𝒫 : 𝑅𝑍 𝒫 Γ— Θ β†’ Qπ‘Ž 𝒫 represents the unobservable reach;

  • β„Žπ‘Žπ‘

𝒫 : π‘…π‘Ž 𝒫 Γ— Ξ£ β†’ Q𝑍 𝒫 represents the observable reach;

  • 𝑧0

= *π‘Ÿ0+ is the initial state.

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Bipartite Dynamic Observer

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

Σ𝑝 = 𝑝 , Σ𝑑 = 𝜏1, 𝜏2 , Σ𝑣𝑝 = *𝑓, 𝑔+ 1 6 4 7 5 𝑝 2

𝜏1

3

𝑝 𝑔 𝑓 𝑝 𝜏2 𝑓, 𝜏1 𝑓

*1,2+, *𝑝, 𝜏1+ *𝑝, 𝜏1+ *1+ A bipartite dynamic observer 𝒫 w.r.t. G is a 7-tuple 𝒫 = (𝑅𝑍

𝒫, π‘…π‘Ž 𝒫, β„Žπ‘π‘Ž 𝒫 , β„Žπ‘Žπ‘ 𝒫 , 𝐹, Ξ“, 𝑧0 )

  • Bipartite Dynamic Observer (BDO)
  • 𝑅𝑍

𝒫 βŠ† 𝐽 is the set of Y-states;

  • π‘…π‘Ž

𝒫 βŠ† 𝐽 Γ— Θ is the set of Z-states so that z = (𝐽 𝑨 , Θ 𝑨 );

  • β„Žπ‘π‘Ž

𝒫 : 𝑅𝑍 𝒫 Γ— Θ β†’ Qπ‘Ž 𝒫 represents the unobservable reach;

  • β„Žπ‘Žπ‘

𝒫 : π‘…π‘Ž 𝒫 Γ— Ξ£ β†’ Q𝑍 𝒫 represents the observable reach;

  • 𝑧0

= *π‘Ÿ0+ is the initial state.

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Bipartite Dynamic Observer

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

Σ𝑝 = 𝑝 , Σ𝑑 = 𝜏1, 𝜏2 , Σ𝑣𝑝 = *𝑓, 𝑔+ 1 6 4 7 5 𝑝 2 3

𝑝 𝑔 𝑓 𝑝 𝜏2 𝑓, 𝜏1 𝑓

*1,2+, *𝑝, 𝜏1+ *4+ 𝜏1 *𝑝, 𝜏1+ *1+

𝜏1

A bipartite dynamic observer 𝒫 w.r.t. G is a 7-tuple 𝒫 = (𝑅𝑍

𝒫, π‘…π‘Ž 𝒫, β„Žπ‘π‘Ž 𝒫 , β„Žπ‘Žπ‘ 𝒫 , 𝐹, Ξ“, 𝑧0 )

  • Bipartite Dynamic Observer (BDO)
  • 𝑅𝑍

𝒫 βŠ† 𝐽 is the set of Y-states;

  • π‘…π‘Ž

𝒫 βŠ† 𝐽 Γ— Θ is the set of Z-states so that z = (𝐽 𝑨 , Θ 𝑨 );

  • β„Žπ‘π‘Ž

𝒫 : 𝑅𝑍 𝒫 Γ— Θ β†’ Qπ‘Ž 𝒫 represents the unobservable reach;

  • β„Žπ‘Žπ‘

𝒫 : π‘…π‘Ž 𝒫 Γ— Ξ£ β†’ Q𝑍 𝒫 represents the observable reach;

  • 𝑧0

= *π‘Ÿ0+ is the initial state.

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9/17

Bipartite Dynamic Observer

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

Σ𝑝 = 𝑝 , Σ𝑑 = 𝜏1, 𝜏2 , Σ𝑣𝑝 = *𝑓, 𝑔+ 1 6 4 7 5 𝑝 2 3

𝑝 𝑔 𝑓 𝑝 𝜏2 𝑓, 𝜏1 𝑓

A bipartite dynamic observer 𝒫 w.r.t. G is a 7-tuple 𝒫 = (𝑅𝑍

𝒫, π‘…π‘Ž 𝒫, β„Žπ‘π‘Ž 𝒫 , β„Žπ‘Žπ‘ 𝒫 , 𝐹, Ξ“, 𝑧0 )

  • Bipartite Dynamic Observer (BDO)
  • 𝑅𝑍

𝒫 βŠ† 𝐽 is the set of Y-states;

  • π‘…π‘Ž

𝒫 βŠ† 𝐽 Γ— Θ is the set of Z-states so that z = (𝐽 𝑨 , Θ 𝑨 );

  • β„Žπ‘π‘Ž

𝒫 : 𝑅𝑍 𝒫 Γ— Θ β†’ Qπ‘Ž 𝒫 represents the unobservable reach;

  • β„Žπ‘Žπ‘

𝒫 : π‘…π‘Ž 𝒫 Γ— Ξ£ β†’ Q𝑍 𝒫 represents the observable reach;

  • 𝑧0

= *π‘Ÿ0+ is the initial state.

*1,2+, *𝑝, 𝜏1+ *4+ 𝜏1 *𝑝, 𝜏1+ *1+ *3+ 𝑝

𝜏1

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Generalized Most Permissive Observer

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

Let G = (Q, Ξ£, πœ€, π‘Ÿ0) be the system and let πœ’: 2𝑅 β†’ *0,1+ be the IS-based property under consideration. The Most Permissive Observer for πœ’ is the BDO β„³π’¬π’«πœ’ = (𝑅𝑍

𝑁𝑄𝑃, π‘…π‘Ž 𝑁𝑄𝑃, β„Žπ‘π‘Ž 𝑁𝑄𝑃, β„Žπ‘Žπ‘ 𝑁𝑄𝑃, Ξ£, Θ, 𝑧0)

defined as the largest BDO

  • 1. For any 𝑧 ∈ 𝑅𝑍

𝑁𝑄𝑃, there exists πœ„ ∈ Θ such that β„Žπ‘π‘Ž 𝑁𝑄𝑃 𝑧, πœ„ !;

  • 2. For any 𝑨 ∈ π‘…π‘Ž

𝑁𝑄𝑃, we have

2.1. βˆ€πœ ∈ Θ 𝑨 (βˆƒπ‘¦ ∈ 𝐽 𝑨 : πœ€ 𝑦, 𝜏 !) β‡’ β„Žπ‘Žπ‘

𝑁𝑄𝑃 𝑨, 𝜏 ! ;

2.2. πœ’ 𝐽 𝑨 = 1

  • Most Permissive Observer (MPO)
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Generalized Most Permissive Observer

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

Let G = (Q, Ξ£, πœ€, π‘Ÿ0) be the system and let πœ’: 2𝑅 β†’ *0,1+ be the IS-based property under consideration. The Most Permissive Observer for πœ’ is the BDO β„³π’¬π’«πœ’ = (𝑅𝑍

𝑁𝑄𝑃, π‘…π‘Ž 𝑁𝑄𝑃, β„Žπ‘π‘Ž 𝑁𝑄𝑃, β„Žπ‘Žπ‘ 𝑁𝑄𝑃, Ξ£, Θ, 𝑧0)

defined as the largest BDO

  • 1. For any 𝑧 ∈ 𝑅𝑍

𝑁𝑄𝑃, there exists πœ„ ∈ Θ such that β„Žπ‘π‘Ž 𝑁𝑄𝑃 𝑧, πœ„ !;

  • 2. For any 𝑨 ∈ π‘…π‘Ž

𝑁𝑄𝑃, we have

2.1. βˆ€πœ ∈ Θ 𝑨 (βˆƒπ‘¦ ∈ 𝐽 𝑨 : πœ€ 𝑦, 𝜏 !) β‡’ β„Žπ‘Žπ‘

𝑁𝑄𝑃 𝑨, 𝜏 ! ;

2.2. πœ’ 𝐽 𝑨 = 1

  • Most Permissive Observer (MPO)
  • MPO for Diagnosability

Cassez, F., & Tripakis, S. (2008). and Dallal, E., & Lafortune, S. (2014).

  • MPO for Opacity

Cassez, F., Dubreil, J., & Marchand, H. (2012).

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Construction of the MPO

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

  • Depth-first search until reach a Z-state violating πœ’
  • Iteratively prune:

Y-state, from which all sensing decisions have been removed Z-state, from which one observation has been removed

*1+ *1,2+, *𝑝, 𝜏1+ *𝑝, 𝜏2+ *4+ 𝜏1 𝜏2 𝑝 *3,7+, *𝑝+ *𝑝+ *6+, *𝑝+ *4+, *𝑝, 𝜏2+ *4,5+, *𝑝+ *1,2,4+, *5+ *6+, *𝑝, 𝜏1+ *5+, *𝑝, 𝜏1+ *𝑝, 𝜏2+ 𝑝 *𝑝+ *𝑝+ *𝑝, 𝜏2+ *𝑝+ *𝑝, 𝜏1+ 𝜏1 *𝑝+ 𝜏2 𝑝 𝑝 𝑝 𝑝 *𝑝+ *1,2,4,5+, *3+ *6+ *3,6+ *𝑝+ 𝑝

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SLIDE 25

12/17

Properties of the MPO

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

There exists a sensor activation policy πœ• such that πœ• ⊨𝐻 πœ’ iff β„³π’¬π’«πœ’ is non-empty.

  • Theorem
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SLIDE 26

12/17

Properties of the MPO

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

πœ• ⊨𝐻 πœ’ iff πœ• is included in β„³π’¬π’«πœ’. There exists a sensor activation policy πœ• such that πœ• ⊨𝐻 πœ’ iff β„³π’¬π’«πœ’ is non-empty.

  • Theorem
  • Theorem
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SLIDE 27

12/17

Properties of the MPO

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

πœ• ⊨𝐻 πœ’ iff πœ• is included in β„³π’¬π’«πœ’. There exists a sensor activation policy πœ• such that πœ• ⊨𝐻 πœ’ iff β„³π’¬π’«πœ’ is non-empty.

  • Theorem
  • Theorem

*1+ *1,2+, *𝑝, 𝜏1+ *𝑝, 𝜏2+ *4+ 𝜏1 𝜏2 𝑝 *3,7+, *𝑝+ *𝑝+ *6+, *𝑝+ *4+, *𝑝, 𝜏2+ *4,5+, *𝑝+ *1,2,4+, *5+ *6+, *𝑝, 𝜏1+ *5+, *𝑝, 𝜏1+ *𝑝, 𝜏2+ 𝑝 *𝑝+ *𝑝+ *𝑝, 𝜏2+ *𝑝+ *𝑝, 𝜏1+ 𝜏1 *𝑝+ 𝜏2 𝑝 𝑝 𝑝 𝑝 *3+ *6+

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SLIDE 28

Synthesis of Optimal Sensor Activation Policies

  • IS-based Sensor Activation Policy

A sensor activation policy πœ• is said to be Information-State-based (or IS-based) if βˆ€π‘‘, 𝑒 ∈ β„’ 𝐻 : β„πœ•

𝑍 𝑑 = β„πœ• 𝑍 𝑒 β‡’ πœ• 𝑑 = πœ•(𝑒)

13/17 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

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SLIDE 29

Synthesis of Optimal Sensor Activation Policies

  • IS-based Sensor Activation Policy

A sensor activation policy πœ• is said to be Information-State-based (or IS-based) if βˆ€π‘‘, 𝑒 ∈ β„’ 𝐻 : β„πœ•

𝑍 𝑑 = β„πœ• 𝑍 𝑒 β‡’ πœ• 𝑑 = πœ•(𝑒)

13/17 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

  • Greedy Optimal Sensor Activation Policy

Suppose that πœ• is a sensor activation policy such that policy πœ• ⊨𝐻 πœ’. We say that πœ• is greedy minimal if βˆ€π‘‘ ∈ β„’ 𝐻 , βˆ€πœ„ ∈ Θ: β„Žπ‘π‘Ž

𝑁𝑄𝑃 β„πœ• 𝑍 𝑑 , πœ„ ! β‡’ πœ„ βŠ„ πœ• 𝑑

The notion of greedy maximality is defined analogously

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SLIDE 30

Synthesis of Optimal Sensor Activation Policies

  • IS-based Sensor Activation Policy

A sensor activation policy πœ• is said to be Information-State-based (or IS-based) if βˆ€π‘‘, 𝑒 ∈ β„’ 𝐻 : β„πœ•

𝑍 𝑑 = β„πœ• 𝑍 𝑒 β‡’ πœ• 𝑑 = πœ•(𝑒)

13/17 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

  • Greedy Optimal Sensor Activation Policy

Suppose that πœ• is a sensor activation policy such that policy πœ• ⊨𝐻 πœ’. We say that πœ• is greedy minimal if βˆ€π‘‘ ∈ β„’ 𝐻 , βˆ€πœ„ ∈ Θ: β„Žπ‘π‘Ž

𝑁𝑄𝑃 β„πœ• 𝑍 𝑑 , πœ„ ! β‡’ πœ„ βŠ„ πœ• 𝑑

The notion of greedy maximality is defined analogously

  • Theorem

Let πœ• be a sensor activation policy such that policy πœ• ⊨𝐻 πœ’. Then πœ• is minimal (respectively, maximal) if it is greedy minimal (respectively, greedy maximal).

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SLIDE 31

Synthesis Procedure

14/17 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

An Optimal 𝝏 s.t. 𝝏 βŠ¨π‘― 𝝌

Property: 𝝌 System: 𝑯

Construct π“π“ π“ŸπŒ Find a greedy minimal IS- based Solution via a DFS

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SLIDE 32

IS-based Formulation of Predictability

  • Theorem

Let πœ’π‘žπ‘ π‘“ be the IS-based predictability defined above. For ant sensor activation policy πœ•, β„’ 𝐻 is predictable w.r.t. 𝑔 and πœ• if and only if πœ• ⊨𝐻 πœ’π‘žπ‘ π‘“.

15/17 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

π’ˆ 𝒕 𝒖 𝒙

𝝐𝑹 π“žπ‘Ή

  • IS-based Predictability πœ’π‘žπ‘ π‘“: 2𝑅 β†’ *0,1+

βˆ€ 𝑗 ∈ 2𝑅: πœ’π‘žπ‘ π‘“ 𝑗 = 0 ⇔ ,βˆƒπ‘Ÿ, π‘Ÿβ€² ∈ 𝑗: π‘Ÿ ∈ πœ–π‘… ∧ π‘Ÿβ€² ∈ π’ͺ

𝑅-

Boundary State [Kumar & Takai, 2010] Non-indicator State

  • Predictability

A live language β„’ 𝐻 is said to be predictable if any fault of the system can be predicted prior to its occurrence with no missed alarm and no false alarm.

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SLIDE 33

Example

16/17 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

1 6 4 7 5

𝑝

2

𝜏1

3

𝑝 π’ˆ 𝑓 𝑝 𝜏2 𝑓, 𝜏1 𝑓

  • Boundary states, πœ–π‘… = 3
  • Non-indicator states, π’ͺ

𝑅 = *1,4,5,6+

  • βˆ€ 𝑗 ∈ 2𝑅: πœ’π‘žπ‘ π‘“ 𝑗 = 1 ⇔ ,βˆ„π‘Ÿ ∈ 1,4,5,6 : 3, π‘Ÿ βŠ† 𝑗-
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SLIDE 34

Example

16/17 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

1 6 4 7 5

𝑝

2

𝜏1

3

𝑝 π’ˆ 𝑓 𝑝 𝜏2 𝑓, 𝜏1 𝑓

  • Boundary states, πœ–π‘… = 3
  • Non-indicator states, π’ͺ

𝑅 = *1,4,5,6+

  • βˆ€ 𝑗 ∈ 2𝑅: πœ’π‘žπ‘ π‘“ 𝑗 = 1 ⇔ ,βˆ„π‘Ÿ ∈ 1,4,5,6 : 3, π‘Ÿ βŠ† 𝑗-

*1+ *1,2+, *𝑝, 𝜏1+ *𝑝, 𝜏2+ *4+ 𝜏1 𝜏2 𝑝 *3,7+, *𝑝+ *𝑝+ *6+, *𝑝+ *4+, *𝑝, 𝜏2+ *4,5+, *𝑝+ *1,2,4+, *5+ *6+, *𝑝, 𝜏1+ *5+, *𝑝, 𝜏1+ *𝑝, 𝜏2+ 𝑝 *𝑝+ *𝑝+ *𝑝, 𝜏2+ *𝑝+ *𝑝, 𝜏1+ 𝜏1 *𝑝+ 𝜏2 𝑝 𝑝 𝑝 𝑝 *3+ *6+

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SLIDE 35

Example

16/17 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

1 6 4 7 5

𝑝

2

𝜏1

3

𝑝 π’ˆ 𝑓 𝑝 𝜏2 𝑓, 𝜏1 𝑓

  • Boundary states, πœ–π‘… = 3
  • Non-indicator states, π’ͺ

𝑅 = *1,4,5,6+

  • βˆ€ 𝑗 ∈ 2𝑅: πœ’π‘žπ‘ π‘“ 𝑗 = 1 ⇔ ,βˆ„π‘Ÿ ∈ 1,4,5,6 : 3, π‘Ÿ βŠ† 𝑗-

*1+ *1,2+, *𝑝, 𝜏1+ *𝑝, 𝜏2+ *4+ 𝜏1 𝜏2 𝑝 *3,7+, *𝑝+ *𝑝+ *6+, *𝑝+ *4+, *𝑝, 𝜏2+ *4,5+, *𝑝+ *1,2,4+, *5+ *6+, *𝑝, 𝜏1+ *5+, *𝑝, 𝜏1+ *𝑝, 𝜏2+ 𝑝 *𝑝+ *𝑝+ *𝑝, 𝜏2+ *𝑝+ *𝑝, 𝜏1+ 𝜏1 *𝑝+ 𝜏2 𝑝 𝑝 𝑝 𝑝 *3+ *6+

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SLIDE 36

Example

16/17 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

1 6 4 7 5

𝑝

2

𝜏1

3

𝑝 π’ˆ 𝑓 𝑝 𝜏2 𝑓, 𝜏1 𝑓

  • Boundary states, πœ–π‘… = 3
  • Non-indicator states, π’ͺ

𝑅 = *1,4,5,6+

  • βˆ€ 𝑗 ∈ 2𝑅: πœ’π‘žπ‘ π‘“ 𝑗 = 1 ⇔ ,βˆ„π‘Ÿ ∈ 1,4,5,6 : 3, π‘Ÿ βŠ† 𝑗-

*1+ *1,2+, *𝑝, 𝜏1+ *𝑝, 𝜏2+ *4+ 𝜏1 𝑝 *3,7+, *𝑝+ *𝑝+ *6+, *𝑝+ *4+, *𝑝, 𝜏2+ *4,5+, *𝑝+ *1,2,4+, *5+ *6+, *𝑝, 𝜏1+ *5+, *𝑝, 𝜏1+ *𝑝, 𝜏2+ 𝑝 *𝑝+ *𝑝+ *𝑝, 𝜏2+ *𝑝+ *𝑝, 𝜏1+ 𝜏1 *𝑝+ 𝜏2 𝑝 𝑝 𝑝 𝑝 *3+ *6+ 𝜏2

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SLIDE 37

Summary

17/17 X.Yin & S.Lafortune (UMich) Dec 2015 CDC 2015

Contributions:

  • A general approach for solving dynamic sensor activation problems
  • Information-state-based properties: A general class of properties
  • Solve previous open problem, e.g., predictability
  • Applicable to more user-defined properties
  • The solution is provably language-based minimal