[PPT] - On Maximal Permissiveness in Partially-Observed Discrete Event PowerPoint Presentation

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Xiang Yin and Stéphane Lafortune

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On Maximal Permissiveness in Partially-Observed Discrete Event Systems: Verification and Synthesis

EECS Department, University of Michigan

13th WODES, May 30-June 1, 2016, Xi’an, China

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

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Introduction

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

Plant G 1 2 3 4 5

Supervisor 𝑇: 𝐹𝑝

∗ → Γ

𝑇(𝑡)

𝑄

Control Engineering Perspective

𝐹 = 𝐹𝑑 ∪

𝐹𝑣𝑑 = 𝐹𝑝 ∪ 𝐹𝑣𝑝

Supervisor: 𝑇: 𝐹𝑝

∗ → 2E; Disable events in 𝐹𝑑 based on its observations

Closed-loop Behavior:𝑀(𝑇/𝐻)

𝑡 𝑄(𝑡)

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Introduction

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

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 initial state
𝐻 = (𝑌, 𝐹, 𝑔, 𝑦0) is a deterministic FSA
Safety specification automaton: 𝑀 𝐼 ⊆ 𝑀 (𝐻)

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Introduction

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

Safe, if 𝑀(𝑇/𝐻) ⊆ 𝑀(𝐼)
Maximally Permissive, if for any safe supervisor 𝑇′, we have 𝑀(𝑇/𝐻) ⊄ 𝑀(𝑇′/𝐻).

We say that a supervisor 𝑇: 𝐹𝑝

∗ → 2𝐹 is

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 initial state
𝐻 = (𝑌, 𝐹, 𝑔, 𝑦0) is a deterministic FSA
Safety specification automaton: 𝑀 𝐼 ⊆ 𝑀 (𝐻)

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Introduction

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

Safe, if 𝑀(𝑇/𝐻) ⊆ 𝑀(𝐼)
Maximally Permissive, if for any safe supervisor 𝑇′, we have 𝑀(𝑇/𝐻) ⊄ 𝑀(𝑇′/𝐻).

We say that a supervisor 𝑇: 𝐹𝑝

∗ → 2𝐹 is

𝑁𝑏𝑦1 𝑁𝑏𝑦2 𝑀(𝐼) 𝑀(𝐻)

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 initial state
𝐻 = (𝑌, 𝐹, 𝑔, 𝑦0) is a deterministic FSA
Safety specification automaton: 𝑀 𝐼 ⊆ 𝑀 (𝐻)

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Literature Review

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

F. Lin, and W. M. Wonham. "On observability of discrete-event systems." Inform. Sci., 44.3 (1988): 173-

198.

R. Cieslak, et al. "Supervisory control of discrete-event processes with partial observations." IEEE

Transactions on Automatic Control, 33.3 (1988): 249-260.

Supremal normal and controllable solution

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Literature Review

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

F. Lin, and W. M. Wonham. "On observability of discrete-event systems." Inform. Sci., 44.3 (1988): 173-

198.

R. Cieslak, et al. "Supervisory control of discrete-event processes with partial observations." IEEE

Transactions on Automatic Control, 33.3 (1988): 249-260.

S. Takai, and T. Ushio. "Effective computation of an ℒ𝑛(𝐻)-closed, controllable, and observable

sublanguage arising in supervisory control." Sys. Cont. Let. 49.3 (2003): 191-200.

K. Cai, R. Zhang, and W. M. Wonham. "Relative observability of discrete-event Systems and its supremal

sublanguages." IEEE Trans. Automatic Control, 60.3 (2015): 659-670.

Supremal normal and controllable solution
Solutions larger than supremal normal and controllable solution

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Literature Review

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

F. Lin, and W. M. Wonham. "On observability of discrete-event systems." Inform. Sci., 44.3 (1988): 173-

198.

R. Cieslak, et al. "Supervisory control of discrete-event processes with partial observations." IEEE

Transactions on Automatic Control, 33.3 (1988): 249-260.

S. Takai, and T. Ushio. "Effective computation of an ℒ𝑛(𝐻)-closed, controllable, and observable

sublanguage arising in supervisory control." Sys. Cont. Let. 49.3 (2003): 191-200.

K. Cai, R. Zhang, and W. M. Wonham. "Relative observability of discrete-event Systems and its supremal

sublanguages." IEEE Trans. Automatic Control, 60.3 (2015): 659-670.

Supremal normal and controllable solution
Solutions larger than supremal normal and controllable solution
These solutions are sound but not complete

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Literature Review

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

F. Lin, and W. M. Wonham. "On observability of discrete-event systems." Inform. Sci., 44.3 (1988): 173-

198.

R. Cieslak, et al. "Supervisory control of discrete-event processes with partial observations." IEEE

Transactions on Automatic Control, 33.3 (1988): 249-260.

S. Takai, and T. Ushio. "Effective computation of an ℒ𝑛(𝐻)-closed, controllable, and observable

sublanguage arising in supervisory control." Sys. Cont. Let. 49.3 (2003): 191-200.

K. Cai, R. Zhang, and W. M. Wonham. "Relative observability of discrete-event Systems and its supremal

sublanguages." IEEE Trans. Automatic Control, 60.3 (2015): 659-670.

Supremal normal and controllable solution
Solutions larger than supremal normal and controllable solution
These solutions are sound but not complete
Solutions are both sound and complete
A certain class of maximal policies
N. Ben Hadj-Alouane, S. Lafortune, and F. Lin. "Centralized and distributed algorithms for on-line

synthesis of maximal control policies under partial observation." Discrete Event Dynamic Systems 6.4 (1996): 379-427.

X. Yin and S. Lafortune. "Synthesis of Maximally Permissive Supervisors for Partially-Observed

Discrete-Event Systems." IEEE Trans. Automatic Control, 61.5 (2016): 1239-1254.

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Literature Review

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

F. Lin, and W. M. Wonham. "On observability of discrete-event systems." Inform. Sci., 44.3 (1988): 173-

198.

R. Cieslak, et al. "Supervisory control of discrete-event processes with partial observations." IEEE

Transactions on Automatic Control, 33.3 (1988): 249-260.

S. Takai, and T. Ushio. "Effective computation of an ℒ𝑛(𝐻)-closed, controllable, and observable

sublanguage arising in supervisory control." Sys. Cont. Let. 49.3 (2003): 191-200.

K. Cai, R. Zhang, and W. M. Wonham. "Relative observability of discrete-event Systems and its supremal

sublanguages." IEEE Trans. Automatic Control, 60.3 (2015): 659-670.

Supremal normal and controllable solution
Solutions larger than supremal normal and controllable solution
These solutions are sound but not complete
Solutions are both sound and complete
A certain class of maximal policies
N. Ben Hadj-Alouane, S. Lafortune, and F. Lin. "Centralized and distributed algorithms for on-line

synthesis of maximal control policies under partial observation." Discrete Event Dynamic Systems 6.4 (1996): 379-427.

X. Yin and S. Lafortune. "Synthesis of Maximally Permissive Supervisors for Partially-Observed

Discrete-Event Systems." IEEE Trans. Automatic Control, 61.5 (2016): 1239-1254.

𝑁𝑏𝑦 𝑀(𝐼)

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Problem Formulation

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

Supervisor Verification Problem.

Given a safe supervisor 𝑇𝑆: 𝐹𝑝

∗ →, 2𝐹, verify whether or not 𝑇𝑆 is maximal.

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Problem Formulation

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

Supervisor Verification Problem.

Given a safe supervisor 𝑇𝑆: 𝐹𝑝

∗ →, 2𝐹, verify whether or not 𝑇𝑆 is maximal.

Supervisor Synthesis Problem.

Given a non-maximal safe supervisor 𝑇𝑆: 𝐹𝑝

∗ → 2𝐹, find a safe supervisor 𝑇

such that 𝑀 𝑇𝑆/𝐻 ⊂ 𝑀 𝑇/𝐻 .

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Problem Formulation

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

Supervisor Verification Problem.

Given a safe supervisor 𝑇𝑆: 𝐹𝑝

∗ →, 2𝐹, verify whether or not 𝑇𝑆 is maximal.

Supervisor Synthesis Problem.

Given a non-maximal safe supervisor 𝑇𝑆: 𝐹𝑝

∗ → 2𝐹, find a safe supervisor 𝑇

such that 𝑀 𝑇𝑆/𝐻 ⊂ 𝑀 𝑇/𝐻 .

Lower bound behavior 𝑀𝑠
𝑴(𝑻𝑺/𝑯) = 𝑴𝒔

↓𝑫𝑷, the infimal controllable and observable super-language

Achieve both the lower bound and permissiveness

Motivation:

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Bipartite Transition System

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

3,6 , *𝑑2+ 4 0 , * + 3 3,5 , *𝑑1+

𝑏 𝑐

*𝑑2+ * + *𝑑1+

𝑼𝑺 1 4 3 5 6 7

𝑏 𝑐 𝑐 𝑏 𝑑1 𝑑1 𝑑2 𝑑2 𝑑2 𝑑1 𝑑1

𝐹𝑑 = 𝑑1, 𝑑2 , 𝐹𝑝 = *𝑏, 𝑐+

Information State: a set of states, 𝐽 ≔ 2𝑌
BTS: A bipartite transition system T w.r.t. G is a 7-tuple

𝑈 = (𝑅𝑍

𝑈, 𝑅𝑎 𝑈, ℎ𝑍𝑎 𝑈 , ℎ𝑎𝑍 𝑈 , 𝐹, Γ, 𝑧0 )

where

𝑅𝑍

𝑈 ⊆ 𝐽 is the set of Y-states;

𝑅𝑎

𝑈 ⊆ 𝐽 × Γ is the set of Z-states so that z = (𝐽 𝑨 , Γ 𝑨 );

ℎ𝑍𝑎

𝑈 : 𝑅𝑍 𝑈 × Γ → Q𝑎 𝑈 represents the unobservable reach;

ℎ𝑎𝑍

𝑈 : 𝑅𝑎 𝑈 × E → Q𝑍 𝑈 represents the observation transition;

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Bipartite Transition System

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

3,6 , *𝑑2+ 4 0 , * + 3 3,5 , *𝑑1+

𝑏 𝑐

*𝑑2+ * + *𝑑1+

𝑼𝑺 1 4 3 5 6 7

𝑏 𝑐 𝑐 𝑏 𝑑1 𝑑1 𝑑2 𝑑2 𝑑2 𝑑1 𝑑1

𝐹𝑑 = 𝑑1, 𝑑2 , 𝐹𝑝 = *𝑏, 𝑐+

The given supervisor 𝑻𝑺 can be realized as a BTS 𝑼𝑺

4 3 5 6

𝑏 𝑐 𝑑1 𝑑2

𝑴(𝑻𝑺/𝑯)

Information State: a set of states, 𝐽 ≔ 2𝑌
BTS: A bipartite transition system T w.r.t. G is a 7-tuple

𝑈 = (𝑅𝑍

𝑈, 𝑅𝑎 𝑈, ℎ𝑍𝑎 𝑈 , ℎ𝑎𝑍 𝑈 , 𝐹, Γ, 𝑧0 )

where

𝑅𝑍

𝑈 ⊆ 𝐽 is the set of Y-states;

𝑅𝑎

𝑈 ⊆ 𝐽 × Γ is the set of Z-states so that z = (𝐽 𝑨 , Γ 𝑨 );

ℎ𝑍𝑎

𝑈 : 𝑅𝑍 𝑈 × Γ → Q𝑎 𝑈 represents the unobservable reach;

ℎ𝑎𝑍

𝑈 : 𝑅𝑎 𝑈 × E → Q𝑍 𝑈 represents the observation transition;

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All Inclusive Controller

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

Definition. (AIC).

The All Inclusive Controller 𝒝ℐ𝒟(𝐻) = (𝑅𝑍

𝐵𝐽𝐷, 𝑅𝑎 𝐵𝐽𝐷, ℎ𝑍𝑎 𝐵𝐽𝐷, ℎ𝑎𝑍 𝐵𝐽𝐷, 𝐹, Γ, 𝑧0),

is defined as the largest BTS such

1. For any 𝑧 ∈ 𝑅𝑍

𝐵𝐽𝐷, there exists at least one control decision

2. For any 𝑨 ∈ 𝑅𝑎

𝐵𝐽𝐷, we have

2.1. all feasible observable events are defined 2.2. 𝐽 𝑨 only contains legal states

* + 3 3 , * + 0,1 , *𝑑1+ 0 , * + 3,5 , *𝑑1+ 3,6 , *𝑑2+ 4,6 , *𝑑2+ 4,5 , *𝑑1+ 3,4,5 , *𝑑1+ 3,4,6 , *𝑑2+

𝑏 𝑐 𝑏, 𝑐

*𝑑1+ *𝑑2+ * + *𝑑1+ *𝑑2+ * + * + *𝑑1+ *𝑑1+ *𝑑2+ 4 4 , * + 3,4 3,4 , * +

𝓑𝓙𝓓 1 4 3 5 6 7

𝑏 𝑐 𝑐 𝑏 𝑑1 𝑑1 𝑑2 𝑑2 𝑑2 𝑑1 𝑑1

𝐹𝑑 = 𝑑1, 𝑑2 , 𝐹𝑝 = *𝑏, 𝑐+

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All Inclusive Controller

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

Definition. (AIC).

The All Inclusive Controller 𝒝ℐ𝒟(𝐻) = (𝑅𝑍

𝐵𝐽𝐷, 𝑅𝑎 𝐵𝐽𝐷, ℎ𝑍𝑎 𝐵𝐽𝐷, ℎ𝑎𝑍 𝐵𝐽𝐷, 𝐹, Γ, 𝑧0),

is defined as the largest BTS such

1. For any 𝑧 ∈ 𝑅𝑍

𝐵𝐽𝐷, there exists at least one control decision

2. For any 𝑨 ∈ 𝑅𝑎

𝐵𝐽𝐷, we have

2.1. all feasible observable events are defined 2.2. 𝐽 𝑨 only contains legal states

* + 3 3 , * + 0,1 , *𝑑1+ 0 , * + 3,5 , *𝑑1+ 3,6 , *𝑑2+ 4,6 , *𝑑2+ 4,5 , *𝑑1+ 3,4,5 , *𝑑1+ 3,4,6 , *𝑑2+

𝑏 𝑐 𝑏, 𝑐

*𝑑1+ *𝑑2+ * + *𝑑1+ *𝑑2+ * + * + *𝑑1+ *𝑑1+ *𝑑2+ 4 4 , * + 3,4 3,4 , * +

𝓑𝓙𝓓 1 4 3 5 6 7

𝑏 𝑐 𝑐 𝑏 𝑑1 𝑑1 𝑑2 𝑑2 𝑑2 𝑑1 𝑑1

𝐹𝑑 = 𝑑1, 𝑑2 , 𝐹𝑝 = *𝑏, 𝑐+

The AIC contains all safe supervisors

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Verification of Maximality: Basic Idea and Difficulties

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

3,6 , *𝑑2+ 4 0 , * + 3 3,5 , *𝑑1+

𝑏 𝑐

*𝑑2+ * + *𝑑1+

𝑼𝑺: realizes the given supervisor

* + 3 3 , * + 0,1 , *𝑑1+ 0 , * + 3,5 , *𝑑1+ 3,6 , *𝑑2+ 4,6 , *𝑑2+ 4,5 , *𝑑1+ 3,4,5 , *𝑑1+ 3,4,6 , *𝑑2+

𝑏 𝑐 𝑏, 𝑐

*𝑑1+ *𝑑2+ * + *𝑑1+ *𝑑2+ * + * + *𝑑1+ *𝑑1+ *𝑑2+ 4 4 , * + 3,4 3,4 , * +

𝓑𝓙𝓓: includes all safe supervisors

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Verification of Maximality: Basic Idea and Difficulties

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

3,6 , *𝑑2+ 4 0 , * + 3 3,5 , *𝑑1+

𝑏 𝑐

*𝑑2+ * + *𝑑1+

𝑼𝑺: realizes the given supervisor

* + 3 3 , * + 0,1 , *𝑑1+ 0 , * + 3,5 , *𝑑1+ 3,6 , *𝑑2+ 4,6 , *𝑑2+ 4,5 , *𝑑1+ 3,4,5 , *𝑑1+ 3,4,6 , *𝑑2+

𝑏 𝑐 𝑏, 𝑐

*𝑑1+ *𝑑2+ * + *𝑑1+ *𝑑2+ * + * + *𝑑1+ *𝑑1+ *𝑑2+ 4 4 , * + 3,4 3,4 , * +

𝓑𝓙𝓓: includes all safe supervisors

“Compare” 𝑼𝑺 with the AIC

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Verification of Maximality: Basic Idea and Difficulties

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

3,6 , *𝑑2+ 4 0 , * + 3 3,5 , *𝑑1+

𝑏 𝑐

*𝑑2+ * + *𝑑1+

𝑼𝑺: realizes the given supervisor

* + 3 3 , * + 0,1 , *𝑑1+ 0 , * + 3,5 , *𝑑1+ 3,6 , *𝑑2+ 4,6 , *𝑑2+ 4,5 , *𝑑1+ 3,4,5 , *𝑑1+ 3,4,6 , *𝑑2+

𝑏 𝑐 𝑏, 𝑐

*𝑑1+ *𝑑2+ * + *𝑑1+ *𝑑2+ * + * + *𝑑1+ *𝑑1+ *𝑑2+ 4 4 , * + 3,4 3,4 , * +

𝓑𝓙𝓓: includes all safe supervisors

“Compare” 𝑼𝑺 with the AIC
How to compare?
The effect of enabling an event depends on future information

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Verification of Maximality: Basic Idea and Difficulties

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

3,6 , *𝑑2+ 4 0 , * + 3 3,5 , *𝑑1+

𝑏 𝑐

*𝑑2+ * + *𝑑1+

𝑼𝑺: realizes the given supervisor

* + 3 3 , * + 0,1 , *𝑑1+ 0 , * + 3,5 , *𝑑1+ 3,6 , *𝑑2+ 4,6 , *𝑑2+ 4,5 , *𝑑1+ 3,4,5 , *𝑑1+ 3,4,6 , *𝑑2+

𝑏 𝑐 𝑏, 𝑐

*𝑑1+ *𝑑2+ * + *𝑑1+ *𝑑2+ * + * + *𝑑1+ *𝑑1+ *𝑑2+ 4 4 , * + 3,4 3,4 , * +

𝓑𝓙𝓓: includes all safe supervisors

B m n k 𝑝 𝑝 𝑑 𝑑

𝑭𝒅 = 𝒅 , 𝑭𝒑 = *𝒑+

3 4 1 𝑝 𝑝 𝑝 𝑝 𝑝 𝑝 𝑑

“Compare” 𝑼𝑺 with the AIC
How to compare?
The effect of enabling an event depends on future information

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Verification of Maximality: Basic Idea and Difficulties

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

3,6 , *𝑑2+ 4 0 , * + 3 3,5 , *𝑑1+

𝑏 𝑐

*𝑑2+ * + *𝑑1+

𝑼𝑺: realizes the given supervisor

* + 3 3 , * + 0,1 , *𝑑1+ 0 , * + 3,5 , *𝑑1+ 3,6 , *𝑑2+ 4,6 , *𝑑2+ 4,5 , *𝑑1+ 3,4,5 , *𝑑1+ 3,4,6 , *𝑑2+

𝑏 𝑐 𝑏, 𝑐

*𝑑1+ *𝑑2+ * + *𝑑1+ *𝑑2+ * + * + *𝑑1+ *𝑑1+ *𝑑2+ 4 4 , * + 3,4 3,4 , * +

𝓑𝓙𝓓: includes all safe supervisors

B m n k 𝑝 𝑝 𝑑 𝑑

𝑭𝒅 = 𝒅 , 𝑭𝒑 = *𝒑+

3 4 1 𝑝 𝑝 𝑝 𝑝 𝑝 𝑝

𝑴(𝑻𝑺/𝑯)

𝑑

“Compare” 𝑼𝑺 with the AIC
How to compare?
The effect of enabling an event depends on future information

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Verification of Maximality: Basic Idea and Difficulties

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

3,6 , *𝑑2+ 4 0 , * + 3 3,5 , *𝑑1+

𝑏 𝑐

*𝑑2+ * + *𝑑1+

𝑼𝑺: realizes the given supervisor

* + 3 3 , * + 0,1 , *𝑑1+ 0 , * + 3,5 , *𝑑1+ 3,6 , *𝑑2+ 4,6 , *𝑑2+ 4,5 , *𝑑1+ 3,4,5 , *𝑑1+ 3,4,6 , *𝑑2+

𝑏 𝑐 𝑏, 𝑐

*𝑑1+ *𝑑2+ * + *𝑑1+ *𝑑2+ * + * + *𝑑1+ *𝑑1+ *𝑑2+ 4 4 , * + 3,4 3,4 , * +

𝓑𝓙𝓓: includes all safe supervisors

B m n k 𝑝 𝑝 𝑑 𝑑

𝑭𝒅 = 𝒅 , 𝑭𝒑 = *𝒑+

3 𝒅 4 1 𝑝 𝑝 𝑝 𝑝 𝑝 𝑝

𝑴(𝑻𝑺/𝑯)

“Compare” 𝑼𝑺 with the AIC
How to compare?
The effect of enabling an event depends on future information

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Verification of Maximality: Basic Idea and Difficulties

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

3,6 , *𝑑2+ 4 0 , * + 3 3,5 , *𝑑1+

𝑏 𝑐

*𝑑2+ * + *𝑑1+

𝑼𝑺: realizes the given supervisor

* + 3 3 , * + 0,1 , *𝑑1+ 0 , * + 3,5 , *𝑑1+ 3,6 , *𝑑2+ 4,6 , *𝑑2+ 4,5 , *𝑑1+ 3,4,5 , *𝑑1+ 3,4,6 , *𝑑2+

𝑏 𝑐 𝑏, 𝑐

*𝑑1+ *𝑑2+ * + *𝑑1+ *𝑑2+ * + * + *𝑑1+ *𝑑1+ *𝑑2+ 4 4 , * + 3,4 3,4 , * +

𝓑𝓙𝓓: includes all safe supervisors

B m n k 𝑝 𝑝 𝑑 𝑑

𝑭𝒅 = 𝒅 , 𝑭𝒑 = *𝒑+

3 𝒅 4 1 𝑝 𝑝 𝑝 𝑝 𝑝 𝑝

𝑴(𝑻𝑺/𝑯)

“Compare” 𝑼𝑺 with the AIC
How to compare?
The effect of enabling an event depends on future information

SLIDE 25

7/14

Verification of Maximality: Basic Idea and Difficulties

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

3,6 , *𝑑2+ 4 0 , * + 3 3,5 , *𝑑1+

𝑏 𝑐

*𝑑2+ * + *𝑑1+

𝑼𝑺: realizes the given supervisor

* + 3 3 , * + 0,1 , *𝑑1+ 0 , * + 3,5 , *𝑑1+ 3,6 , *𝑑2+ 4,6 , *𝑑2+ 4,5 , *𝑑1+ 3,4,5 , *𝑑1+ 3,4,6 , *𝑑2+

𝑏 𝑐 𝑏, 𝑐

*𝑑1+ *𝑑2+ * + *𝑑1+ *𝑑2+ * + * + *𝑑1+ *𝑑1+ *𝑑2+ 4 4 , * + 3,4 3,4 , * +

𝓑𝓙𝓓: includes all safe supervisors

B m n k 𝑝 𝑝 𝑑 𝑑

𝑭𝒅 = 𝒅 , 𝑭𝒑 = *𝒑+

3 𝒅 4 1 𝑝 𝑝 𝑝 𝑝 𝑝 𝑝

𝑴(𝑻𝑺/𝑯) Conflict!

“Compare” 𝑼𝑺 with the AIC
How to compare?
The effect of enabling an event depends on future information

SLIDE 26

8/14

Control Simulation Relation

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

Let 𝑈

1 and 𝑈 2 between BTSs. A relation Φ = Φ𝑍 ∪ Φ𝑎 ⊆ 𝑅𝑍 𝑈

1 × 𝑅𝑍

𝑈

2 × (𝑅𝑎

𝑈

1 × 𝑅𝑎

𝑈

2)

is said to be a control simulation relation from 𝑼𝟐 to 𝑼𝟑 if the following conditions hold: 1. y0, y0 ∈ ΦY; 2. For every y1, y2 ∈ ΦY, we have that: for any y1

γ1

→ z1in T

1, there exists y2 γ2

→ z2 such that z1, z2 ∈ ΦZ and 𝛅𝟐 ⊆ 𝛅𝟑. 3. For every z1, z2 ∈ Φz, we have that: for any z1

σ

→ y1in T

1, there exists

z2

σ

→ y2such that y1, y2 ∈ ΦY.

Definition. (Control Simulation Relation)

SLIDE 27

8/14

Control Simulation Relation

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

Let 𝑈

1 and 𝑈 2 between BTSs. A relation Φ = Φ𝑍 ∪ Φ𝑎 ⊆ 𝑅𝑍 𝑈

1 × 𝑅𝑍

𝑈

2 × (𝑅𝑎

𝑈

1 × 𝑅𝑎

𝑈

2)

is said to be a control simulation relation from 𝑼𝟐 to 𝑼𝟑 if the following conditions hold: 1. y0, y0 ∈ ΦY; 2. For every y1, y2 ∈ ΦY, we have that: for any y1

γ1

→ z1in T

1, there exists y2 γ2

→ z2 such that z1, z2 ∈ ΦZ and 𝛅𝟐 ⊆ 𝛅𝟑. 3. For every z1, z2 ∈ Φz, we have that: for any z1

σ

→ y1in T

1, there exists

z2

σ

→ y2such that y1, y2 ∈ ΦY.

Definition. (Control Simulation Relation)
There exists a unique maximal CSR Φ∗ 𝑈

1, 𝑈2 from 𝑈 1to 𝑈 2 if one exists

Φ∗ 𝑈

1, 𝑈 2 can be computed by

Φ∗ 𝑈

1, 𝑈 2 = lim 𝑙→∞ 𝐺𝑙((𝑅𝑍 𝑈

1 × 𝑅𝑍

𝑈

2) ∪ (𝑅𝑎

𝑈

1 × 𝑅𝑎

𝑈

2))

SLIDE 28

9/14

Verification of Maximality: Solution

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

3,6 , *𝑑2+ 4 0 , * + 3 3,5 , *𝑑1+

𝑏 𝑐

*𝑑2+ * + *𝑑1+

𝑼𝑺

* + 3 3 , * + 0,1 , *𝑑1+ 0 , * + 3,5 , *𝑑1+ 3,6 , *𝑑2+ 4,6 , *𝑑2+ 4,5 , *𝑑1+ 3,4,5 , *𝑑1+ 3,4,6 , *𝑑2+

𝑏 𝑐 𝑏, 𝑐

*𝑑1+ *𝑑2+ * + *𝑑1+ *𝑑2+ * + * + *𝑑1+ *𝑑1+ *𝑑2+ 4 4 , * + 3,4 3,4 , * +

𝓑𝓙𝓓

𝜲∗

SLIDE 29

9/14

Verification of Maximality: Solution

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

𝑼𝑺 𝓑𝓙𝓓

Let 𝑧 be a Y-state in 𝑈

𝑆 and 𝑑𝑈𝑆 𝑧 be the control decision defined at 𝑧.

We say that control decision 𝛿 replaces 𝑑𝑈𝑆 𝑧 at 𝑧 if

1. 𝛿 is defined at 𝑧 in the AIC
2. 𝑑𝑈𝑆 𝑧 ⊂ 𝛿
3. 𝑨, 𝑨′ ∈ Φ∗(𝑈

𝑆, 𝒝ℐ𝒟), where y 𝑑𝑈𝑆(𝑧)

𝑨 and y

𝛿

→ 𝑨′ Replacement.

3,6 , *𝑑2+ 4 0 , * + 3 3,5 , *𝑑1+

𝑏 𝑐

*𝑑2+ * + *𝑑1+ * + 3 3 , * + 0,1 , *𝑑1+ 0 , * + 3,5 , *𝑑1+ 3,6 , *𝑑2+ 4,6 , *𝑑2+ 4,5 , *𝑑1+ 3,4,5 , *𝑑1+ 3,4,6 , *𝑑2+

𝑏 𝑐 𝑏, 𝑐

*𝑑1+ *𝑑2+ * + *𝑑1+ *𝑑2+ * + * + *𝒅𝟐+ *𝑑1+ *𝑑2+ 4 4 , * + 3,4 3,4 , * +

𝜲∗

SLIDE 30

9/14

Verification of Maximality: Solution

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

Let 𝑧 be a Y-state in 𝑈

𝑆 and 𝑑𝑈𝑆 𝑧 be the control decision defined at 𝑧.

We say that control decision 𝛿 replaces 𝑑𝑈𝑆 𝑧 at 𝑧 if

1. 𝛿 is defined at 𝑧 in the AIC
2. 𝑑𝑈𝑆 𝑧 ⊂ 𝛿
3. 𝑨, 𝑨′ ∈ Φ∗(𝑈

𝑆, 𝒝ℐ𝒟), where y 𝑑𝑈𝑆(𝑧)

𝑨 and y

𝛿

→ 𝑨′ Replacement. Supervisor 𝑇𝑆 is maximal iff no control decision in 𝑈

𝑆 can be replaced

Theorem.

𝑼𝑺

3,6 , *𝑑2+ 4 0 , * + 3 3,5 , *𝑑1+

𝑏 𝑐

*𝑑2+ * + *𝑑1+ * + 3 3 , * + 0,1 , *𝑑1+ 0 , * + 3,5 , *𝑑1+ 3,6 , *𝑑2+ 4,6 , *𝑑2+ 4,5 , *𝑑1+ 3,4,5 , *𝑑1+ 3,4,6 , *𝑑2+

𝑏 𝑐 𝑏, 𝑐

*𝑑1+ *𝑑2+ * + *𝑑1+ *𝑑2+ * + * + *𝒅𝟐+ *𝑑1+ *𝑑2+ 4 4 , * + 3,4 3,4 , * +

𝜲∗

𝓑𝓙𝓓

SLIDE 31

10/14

Synthesis of Larger Supervisor: Basic Idea and Difficulties

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

Construct a new BTS: replace the original control decision in 𝑼𝑺 by a larger one

SLIDE 32

10/14

Synthesis of Larger Supervisor: Basic Idea and Difficulties

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

Construct a new BTS: replace the original control decision in 𝑼𝑺 by a larger one
Such a BTS may not exist!

SLIDE 33

10/14

Synthesis of Larger Supervisor: Basic Idea and Difficulties

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

Construct a new BTS: replace the original control decision in 𝑼𝑺 by a larger one
Such a BTS may not exist!

3,6 , *𝑑2+ 4 0 , * + 3 3,5 , *𝑑1+

𝑏 𝑐

*𝑑2+ *𝑑1+

𝑼𝑺

* + 3 3 , * + 0,1 , *𝑑1+ 0 , * + 3,5 , *𝑑1+ 3,6 , *𝑑2+ 4,6 , *𝑑2+ 4,5 , *𝑑1+ 3,4,5 , *𝑑1+ 3,4,6 , *𝑑2+

𝑏 𝑐 𝑏, 𝑐

*𝑑1+ *𝑑2+ * + *𝑑1+ *𝑑2+ * + * + *𝒅𝟐+ *𝑑1+ *𝑑2+ 4 4 , * + 3,4 3,4 , * +

𝓑𝓙𝓓

𝜲∗

* +

SLIDE 34

10/14

Synthesis of Larger Supervisor: Basic Idea and Difficulties

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

Construct a new BTS: replace the original control decision in 𝑼𝑺 by a larger one
Such a BTS may not exist!

3,6 , *𝑑2+ 4 0 , * + 3 3,5 , *𝑑1+

𝑏 𝑐

*𝑑2+ *𝑑1+

𝑼𝑺

* + 3 3 , * + 0,1 , *𝑑1+ 0 , * + 3,5 , *𝑑1+ 3,6 , *𝑑2+ 4,6 , *𝑑2+ 4,5 , *𝑑1+ 3,4,5 , *𝑑1+ 3,4,6 , *𝑑2+

𝑏 𝑐 𝑏, 𝑐

*𝑑1+ *𝑑2+ * + *𝑑1+ *𝑑2+ * + * + *𝒅𝟐+ *𝑑1+ *𝑑2+ 4 4 , * + 3,4 3,4 , * +

𝓑𝓙𝓓

𝜲∗

* +

SLIDE 35

10/14

Synthesis of Larger Supervisor: Basic Idea and Difficulties

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

Construct a new BTS: replace the original control decision in 𝑼𝑺 by a larger one
Such a BTS may not exist!

3,6 , *𝑑2+ 4 0 , * + 3 3,5 , *𝑑1+

𝑏 𝑐

*𝑑2+ *𝑑1+

𝑼𝑺

* + 3 3 , * + 0,1 , *𝑑1+ 0 , * + 3,5 , *𝑑1+ 3,6 , *𝑑2+ 4,6 , *𝑑2+ 4,5 , *𝑑1+ 3,4,5 , *𝑑1+ 3,4,6 , *𝑑2+

𝑏 𝑐 𝑏, 𝑐

*𝑑1+ *𝑑2+ * + *𝑑1+ *𝑑2+ * + * + *𝒅𝟐+ *𝑑1+ *𝑑2+ 4 4 , * + 3,4 3,4 , * +

𝓑𝓙𝓓

𝜲∗

* +

SLIDE 36

10/14

Synthesis of Larger Supervisor: Basic Idea and Difficulties

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

Construct a new BTS: replace the original control decision in 𝑼𝑺 by a larger one
Such a BTS may not exist!
Information Merge Phenomenon: Information is lost

3,6 , *𝑑2+ 4 0 , * + 3 3,5 , *𝑑1+

𝑏 𝑐

*𝑑2+ *𝑑1+

𝑼𝑺

* + 3 3 , * + 0,1 , *𝑑1+ 0 , * + 3,5 , *𝑑1+ 3,6 , *𝑑2+ 4,6 , *𝑑2+ 4,5 , *𝑑1+ 3,4,5 , *𝑑1+ 3,4,6 , *𝑑2+

𝑏 𝑐 𝑏, 𝑐

*𝑑1+ *𝑑2+ * + *𝑑1+ *𝑑2+ * + * + *𝒅𝟐+ *𝑑1+ *𝑑2+ 4 4 , * + 3,4 3,4 , * +

𝓑𝓙𝓓

𝜲∗

* +

SLIDE 37

10/14

Synthesis of Larger Supervisor: Basic Idea and Difficulties

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

Construct a new BTS: replace the original control decision in 𝑼𝑺 by a larger one
Such a BTS may not exist!
Information Merge Phenomenon: Information is lost
𝟑𝒀 may not be sufficient for the synthesis problem!

3,6 , *𝑑2+ 4 0 , * + 3 3,5 , *𝑑1+

𝑏 𝑐

*𝑑2+ *𝑑1+

𝑼𝑺

* + 3 3 , * + 0,1 , *𝑑1+ 0 , * + 3,5 , *𝑑1+ 3,6 , *𝑑2+ 4,6 , *𝑑2+ 4,5 , *𝑑1+ 3,4,5 , *𝑑1+ 3,4,6 , *𝑑2+

𝑏 𝑐 𝑏, 𝑐

*𝑑1+ *𝑑2+ * + *𝑑1+ *𝑑2+ * + * + *𝒅𝟐+ *𝑑1+ *𝑑2+ 4 4 , * + 3,4 3,4 , * +

𝓑𝓙𝓓

𝜲∗

* +

SLIDE 38

11/14

The Role of Strict Sub-automaton

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

The information merge phenomenon will not occur under the assumption

that 𝑆 is a strict sub-automaton of 𝐻, where 𝑀 𝑆 = 𝑀(𝑇𝑆/𝐻).

1 𝑏 𝑐

1 𝑏

1’ Sub-automaton strict sub-automaton

𝑯𝟐 𝑯𝟑 𝑯 𝑐

SLIDE 39

11/14

The Role of Strict Sub-automaton

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

The information merge phenomenon will not occur under the assumption

that 𝑆 is a strict sub-automaton of 𝐻, where 𝑀 𝑆 = 𝑀(𝑇𝑆/𝐻).

Strict sub-automaton: if a string goes outside, then it stays outside forever

1 𝑏 𝑐

1 𝑏

1’ Sub-automaton strict sub-automaton

𝑯𝟐 𝑯𝟑 𝑯 𝑐

SLIDE 40

11/14

The Role of Strict Sub-automaton

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

The information merge phenomenon will not occur under the assumption

that 𝑆 is a strict sub-automaton of 𝐻, where 𝑀 𝑆 = 𝑀(𝑇𝑆/𝐻).

Strict sub-automaton: if a string goes outside, then it stays outside forever
We can always obtain strict sub-automaton [Cho & Marcus, 1989]

1 𝑏

1’ Sub-automaton strict sub-automaton

𝑯𝟑 𝑯 𝑐

1 𝑏 𝑐 𝑯𝟐

SLIDE 41

11/14

The Role of Strict Sub-automaton

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

4 3 5 6

𝑏 𝑐 𝑑1 𝑑2

1 4 3 5 6 7

𝑏 𝑐 𝑐 𝑏 𝑑1 𝑑1 𝑑2 𝑑2 𝑑2 𝑑1 𝑑1

Sub-automaton 𝑺 𝑯

SLIDE 42

11/14

The Role of Strict Sub-automaton

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

4 3 5 6

𝑏 𝑐 𝑑1 𝑑2

1 4 3 5 6 7

𝑏 𝑐 𝑐 𝑏 𝑑1 𝑑1 𝑑2 𝑑2 𝑑2 𝑑1 𝑑1

1 4 3 5 6 7

𝑏 𝑐 𝑐 𝑏 𝑑1 𝑑1 𝑑2 𝑑2 𝑑2 𝑑1 𝑑1

3 ’ 4 ’ 5 6 ’ ’

𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1

Refine 𝑺 𝑯 𝑯′

SLIDE 43

11/14

The Role of Strict Sub-automaton

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

4 3 5 6

𝑏 𝑐 𝑑1 𝑑2

1 4 3 5 6 7

𝑏 𝑐 𝑐 𝑏 𝑑1 𝑑1 𝑑2 𝑑2 𝑑2 𝑑1 𝑑1

1 4 3 5 6 7

𝑏 𝑐 𝑐 𝑏 𝑑1 𝑑1 𝑑2 𝑑2 𝑑2 𝑑1 𝑑1

3 ’ 4 ’ 5 6 ’ ’

𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1

Refine Strict sub-automaton 𝑺 𝑯 𝑯′

SLIDE 44

11/14

The Role of Strict Sub-automaton

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

4 3 5 6

𝑏 𝑐 𝑑1 𝑑2

1 4 3 5 6 7

𝑏 𝑐 𝑐 𝑏 𝑑1 𝑑1 𝑑2 𝑑2 𝑑2 𝑑1 𝑑1

1 4 3 5 6 7

𝑏 𝑐 𝑐 𝑏 𝑑1 𝑑1 𝑑2 𝑑2 𝑑2 𝑑1 𝑑1

3 ’ 4 ’ 5 6 ’ ’

𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1

Refine Strict sub-automaton 𝑺 𝑯 𝑯′

* + 3 3 , * + 0 , * + 3,5 , *𝑑1+ 3,6′ , *𝑑2+ 4,6 , *𝑑2+ 4,5′ , *𝑑1+ 3′, 4,5′ , *𝑑1+ 3′, 4,6,6′ , *𝑑2+

𝑏 𝑐 𝑐

*𝑑1+ *𝑑2+ *𝑑1+ *𝑑2+ * + * + *𝑑1+ * + *𝑑1+ *𝑑2+ 4 4 , * + 3′, 4 3′, 4 , * + 3,4′, 5,5′ , *𝑑1+ 3,4′, 6′ , *𝑑2+ *𝑑1+ *𝑑2+ * +

𝑏

0,1 , *𝑑1+ 3,4′ , * + 3,4′

AIC for the refined system

SLIDE 45

11/14

The Role of Strict Sub-automaton

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

4 3 5 6

𝑏 𝑐 𝑑1 𝑑2

1 4 3 5 6 7

𝑏 𝑐 𝑐 𝑏 𝑑1 𝑑1 𝑑2 𝑑2 𝑑2 𝑑1 𝑑1

1 4 3 5 6 7

𝑏 𝑐 𝑐 𝑏 𝑑1 𝑑1 𝑑2 𝑑2 𝑑2 𝑑1 𝑑1

3 ’ 4 ’ 5 6 ’ ’

𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1

Refine Strict sub-automaton 𝑺 𝑯 𝑯′

* + 3 3 , * + 0 , * + 3,5 , *𝑑1+ 3,6′ , *𝑑2+ 4,6 , *𝑑2+ 4,5′ , *𝑑1+ 3′, 4,5′ , *𝑑1+ 3′, 4,6,6′ , *𝑑2+

𝑏 𝑐 𝑐

*𝑑1+ *𝑑2+ *𝑑1+ *𝑑2+ * + * + *𝑑1+ * + *𝑑1+ *𝑑2+ 4 4 , * + 3′, 4 3′, 4 , * + 3,4′, 5,5′ , *𝑑1+ 3,4′, 6′ , *𝑑2+ *𝑑1+ *𝑑2+ * +

𝑏

0,1 , *𝑑1+ 3,4′ , * + 3,4′

AIC for the refined system

SLIDE 46

12/14

Synthesis Algorithm

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

1. Construct BTS 𝑈

𝑆 and 𝒝ℐ𝒟(𝐻) (make sure 𝑆 is a strict sub-automaton of 𝐻)

2. Compute the maximal CSR Φ∗ from 𝑈

𝑆 to 𝒝ℐ𝒟(𝐻)

3. Find a Y-state 𝑧 in 𝑈

𝑆 such that its control decision can be replaced by 𝛿

4. Construct BTS 𝑈∗ by

For Y-state 𝑧, choose 𝛿 which is larger than 𝑑𝑈𝑆(𝑧)
For other Y-states, choose the same control decisions in 𝑈

𝑆

Synthesis Steps:

SLIDE 47

12/14

Synthesis Algorithm

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

1. Construct BTS 𝑈

𝑆 and 𝒝ℐ𝒟(𝐻) (make sure 𝑆 is a strict sub-automaton of 𝐻)

2. Compute the maximal CSR Φ∗ from 𝑈

𝑆 to 𝒝ℐ𝒟(𝐻)

3. Find a Y-state 𝑧 in 𝑈

𝑆 such that its control decision can be replaced by 𝛿

4. Construct BTS 𝑈∗ by

For Y-state 𝑧, choose 𝛿 which is larger than 𝑑𝑈𝑆(𝑧)
For other Y-states, choose the same control decisions in 𝑈

𝑆

Synthesis Steps:

𝑼𝑺 𝓑𝓙𝓓

* + 3

𝑏 𝑐 𝑐

*𝑑1+ *𝑑2+ *𝑑2+ * + * + *𝑑1+ * + *𝑑1+ *𝑑2+ *𝑑1+ *𝑑2+ * +

𝑏

*𝑑1+ 4

𝑏 𝑐

*𝑑2+ * + *𝑑1+ 4 3 3′, 4 3,4′

SLIDE 48

12/14

Synthesis Algorithm

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

1. Construct BTS 𝑈

𝑆 and 𝒝ℐ𝒟(𝐻) (make sure 𝑆 is a strict sub-automaton of 𝐻)

2. Compute the maximal CSR Φ∗ from 𝑈

𝑆 to 𝒝ℐ𝒟(𝐻)

3. Find a Y-state 𝑧 in 𝑈

𝑆 such that its control decision can be replaced by 𝛿

4. Construct BTS 𝑈∗ by

For Y-state 𝑧, choose 𝛿 which is larger than 𝑑𝑈𝑆(𝑧)
For other Y-states, choose the same control decisions in 𝑈

𝑆

Synthesis Steps:

𝑼𝑺 𝓑𝓙𝓓

* + 3

𝑏 𝑐 𝑐

*𝑑1+ *𝑑2+ *𝑑2+ * + * + *𝑑1+ * + *𝑑1+ *𝑑2+ *𝑑1+ *𝑑2+ * +

𝑏

*𝑑1+ 4

𝑏 𝑐

*𝑑2+ * + *𝑑1+ 4 3 3′, 4 3,4′

SLIDE 49

12/14

Synthesis Algorithm

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

1. Construct BTS 𝑈

𝑆 and 𝒝ℐ𝒟(𝐻) (make sure 𝑆 is a strict sub-automaton of 𝐻)

2. Compute the maximal CSR Φ∗ from 𝑈

𝑆 to 𝒝ℐ𝒟(𝐻)

3. Find a Y-state 𝑧 in 𝑈

𝑆 such that its control decision can be replaced by 𝛿

4. Construct BTS 𝑈∗ by

For Y-state 𝑧, choose 𝛿 which is larger than 𝑑𝑈𝑆(𝑧)
For other Y-states, choose the same control decisions in 𝑈

𝑆

Synthesis Steps:

𝑼𝑺 𝓑𝓙𝓓

* + 3

𝑏 𝑐 𝑐

*𝑑1+ *𝑑2+ *𝑑2+ * + * + *𝑑1+ * + *𝑑1+ *𝑑2+ *𝑑1+ *𝑑2+ * +

𝑏

*𝑑1+ 4

𝑏 𝑐

*𝑑2+ * + *𝑑1+ 4 3 3′, 4 3,4′

SLIDE 50

12/14

Synthesis Algorithm

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

1. Construct BTS 𝑈

𝑆 and 𝒝ℐ𝒟(𝐻) (make sure 𝑆 is a strict sub-automaton of 𝐻)

2. Compute the maximal CSR Φ∗ from 𝑈

𝑆 to 𝒝ℐ𝒟(𝐻)

3. Find a Y-state 𝑧 in 𝑈

𝑆 such that its control decision can be replaced by 𝛿

4. Construct BTS 𝑈∗ by

For Y-state 𝑧, choose 𝛿 which is larger than 𝑑𝑈𝑆(𝑧)
For other Y-states, choose the same control decisions in 𝑈

𝑆

Synthesis Steps:

𝑼𝑺 𝓑𝓙𝓓

* + 3

𝑏 𝑐 𝑐

*𝑑1+ *𝑑2+ *𝑑2+ * + * + *𝑑1+ * + *𝑑1+ *𝑑2+ *𝑑1+ *𝑑2+ * +

𝑏

*𝑑1+ 4

𝑏 𝑐

*𝑑2+ * + *𝑑1+ 4 3 3′, 4 3,4′

SLIDE 51

12/14

Synthesis Algorithm

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

1. Construct BTS 𝑈

𝑆 and 𝒝ℐ𝒟(𝐻) (make sure 𝑆 is a strict sub-automaton of 𝐻)

2. Compute the maximal CSR Φ∗ from 𝑈

𝑆 to 𝒝ℐ𝒟(𝐻)

3. Find a Y-state 𝑧 in 𝑈

𝑆 such that its control decision can be replaced by 𝛿

4. Construct BTS 𝑈∗ by

For Y-state 𝑧, choose 𝛿 which is larger than 𝑑𝑈𝑆(𝑧)
For other Y-states, choose the same control decisions in 𝑈

𝑆

Synthesis Steps:

𝑼𝑺 𝓑𝓙𝓓

* + 3

𝑏 𝑐 𝑐

*𝑑1+ *𝑑2+ *𝑑2+ * + * + *𝑑1+ * + *𝑑1+ *𝑑2+ *𝑑1+ *𝑑2+ * +

𝑏

*𝑑1+ 4

𝑏 𝑐

*𝑑2+ * + *𝑑1+ 4 3 3′, 4 3,4′

𝟑𝒀 is sufficient!

SLIDE 52

13/14

Achieve More Permissiveness

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

No

Find a supervisor 𝑇𝑆′ strictly larger than 𝑇𝑆 𝑈𝑆 and AIC Is 𝑼𝑺 maximal?

Refine the state-space

f 𝑆 s.t. 𝑆 ⊏ 𝐻

Yes

A maximal supervisor 𝑻𝑺 ← 𝑻𝑺

′

𝑴 𝑺 ← 𝑴(𝑻𝑺/𝑯)

SLIDE 53

13/14

Achieve More Permissiveness

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

No

Find a supervisor 𝑇𝑆′ strictly larger than 𝑇𝑆 𝑈𝑆 and AIC Is 𝑼𝑺 maximal?

Refine the state-space

f 𝑆 s.t. 𝑆 ⊏ 𝐻

Yes

A maximal supervisor 𝑻𝑺 ← 𝑻𝑺

′

𝑴 𝑺 ← 𝑴(𝑻𝑺/𝑯)

Iteration may not converge!

SLIDE 54

14/14

Conclusion

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

Contribution

Verify whether or not a given supervisor is maximal
The notion of control simulation relation
Synthesis a new supervisor that is strictly more permissive
Information Merge Phenomenon & Strict sub-automaton

SLIDE 55

14/14

Conclusion

X.Yin & S.Lafortune (UMich) May 2016 WODES 2016

Contribution

Verify whether or not a given supervisor is maximal
The notion of control simulation relation
Synthesis a new supervisor that is strictly more permissive
Information merge phenomenon & Strict sub-automaton

On Going Work

Synthesize a maximal supervisor that contains the given one