A General Approach for Synthesis of Supervisors for - - PowerPoint PPT Presentation

β–Ά
a general approach for synthesis of supervisors for
SMART_READER_LITE
LIVE PREVIEW

A General Approach for Synthesis of Supervisors for - - PowerPoint PPT Presentation

Oct 20, 2022 536 likes β€’1.03k views

A General Approach for Synthesis of Supervisors for Partially-Observed Discrete-Event Systems Xiang Yin and Stphane Lafortune EECS Department, University of Michigan 19th IFAC WC, August 24-29, 2014, Cape Town, South Africa 1/18 X.Yin &

slide-1
SLIDE 1

Xiang Yin and StΓ©phane Lafortune

1/18

A General Approach for Synthesis of Supervisors for Partially-Observed Discrete-Event Systems

EECS Department, University of Michigan

19th IFAC WC, August 24-29, 2014, Cape Town, South Africa

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

slide-2
SLIDE 2

2/18

Introduction

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

Plant G 1 2 3 4 5

Supervisor 𝑇: 𝐹𝑝

βˆ— β†’ Ξ“

𝑇(𝑑) 𝑑 𝑄(𝑑)

𝑄

  • Supervisor 𝑇: 𝑄 β„’ 𝐻

β†’ Ξ“, where Ξ“ ≔ {𝛿 ∈ 2𝐹: 𝐹𝑣𝑑 βŠ† 𝛿}

  • Supervisory control under partial observation
  • 𝐹 = 𝐹𝑑

βˆͺ 𝐹𝑣𝑑 = 𝐹𝑝 βˆͺ 𝐹𝑣𝑝

slide-3
SLIDE 3

3/18

System Model

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

𝐻 = (π‘Œ, 𝐹, 𝑔, 𝑦0) is a deterministic FSA

  • π‘Œ is the finite set of states;
  • 𝐹 is the finite set of events;
  • 𝑔: π‘Œ Γ— 𝐹 β†’ π‘Œ is the partial transition function;
  • 𝑦0 is the initial state.
slide-4
SLIDE 4

3/18

System Model

𝐻 = (π‘Œ, 𝐹, 𝑔, 𝑦0) is a deterministic FSA

  • π‘Œ is the finite set of states;
  • 𝐹 is the finite set of events;
  • 𝑔: π‘Œ Γ— 𝐹 β†’ π‘Œ is the partial transition function;
  • 𝑦0 is the initial state.
  • Specification automaton 𝐼: 𝐿 = β„’ 𝐼 βŠ† β„’ (𝐻)
  • Assumption : illegality is captured by states (w.l.o.g.)

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

π‘ŒπΌ βŠ† π‘Œ is the set of legal states

slide-5
SLIDE 5

4/18

Problem Formulation

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

  • Existence Condition: (Controllability and Observability Theorem)

There exists a supervisor such that β„’ (𝑇/𝐻) = 𝐿 if and only if 𝐿 is controllable and observable.

slide-6
SLIDE 6

4/18

Problem Formulation

  • Synthesis Problem: (BSCOP 𝑛𝑏𝑦)
  • Existence Condition: (Controllability and Observability Theorem)

Given a plant 𝐻 and specification 𝐼. Find a supervisor 𝑇: 𝐹𝑝

βˆ— β†’ Ξ“ such that

1). β„’ (𝑇/𝐻) βŠ† β„’ 𝐼 ; (Safety) 2). β„’(𝑇/𝐻) βŠ„ β„’(𝑇′/𝐻), βˆ€ safe 𝑇′. (Maximal Permissiveness) There exists a supervisor such that β„’ (𝑇/𝐻) = 𝐿 if and only if 𝐿 is controllable and observable.

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

slide-7
SLIDE 7

5/18

Literature Survey

  • F. Lin, and W. M. Wonham. "On observability of discrete-event systems." Information

sciences 44.3 (1988): 173-198.

  • R. Cieslak, et al. "Supervisory control of discrete-event processes with partial
  • bservations." IEEE Transactions on Automatic Control, 33.3 (1988): 249-260.

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

(Initial works; Supremal normal solution)

slide-8
SLIDE 8

5/18

Literature Survey

  • F. Lin, and W. M. Wonham. "On observability of discrete-event systems." Information

sciences 44.3 (1988): 173-198.

  • R. Cieslak, et al. "Supervisory control of discrete-event processes with partial
  • bservations." IEEE Transactions on Automatic Control, 33.3 (1988): 249-260.
  • K. Cai, R. Zhang, and W. M. Wonham. "Relative observability of discrete-event

Systems and its supremal sublanguages." IEEE Transactions on Automatic Control, (2014).

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

and observable sublanguage arising in supervisory control." Systems & Control Letters 49.3 (2003): 191-200.

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

(Initial works; Supremal normal solution) (Solutions larger than supremal normal)

slide-9
SLIDE 9

6/18

Literature Survey

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

  • Hadj-Alouane, Nejib Ben, StΓ©phane Lafortune, and Feng 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.

  • Heymann, Michael, and Feng Lin. "On-line control of partially observed discrete

event systems." Discrete Event Dynamic Systems 4.3 (1994): 221-236. (Online control; Only for safety specification; A certain class of maximal policies)

slide-10
SLIDE 10

6/18

Literature Survey

  • T.-S. Yoo, and S. Lafortune. "Solvability of centralized supervisory control under

partial observation." Discrete Event Dynamic Systems 16.4 (2006): 527-553.

  • K. Inan, β€œNondeterministic supervision under partial observations,” in 11th

International Conference on Analysis and Optimization of Systems: Discrete Event

  • Systems. Springer, (1994): 39–48.

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

(Decidability for safe and non-blocking; No synthesis) (Solvability for safe and non-blocking; No maximality)

  • Hadj-Alouane, Nejib Ben, StΓ©phane Lafortune, and Feng 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.

  • Heymann, Michael, and Feng Lin. "On-line control of partially observed discrete

event systems." Discrete Event Dynamic Systems 4.3 (1994): 221-236. (Online control; Only for safety specification; A certain class of maximal policies)

slide-11
SLIDE 11

7/18

The Need for a New Approach

 Observability is not preserved under union

  • algebraic approach cannot obtain a maximal solution
  • synthesis of maximally-permissive safe and non-blocking

supervisor is open

 Solution space may be infinite

Why we need a new approach?

  • how to solve optimal control problem?

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

slide-12
SLIDE 12

8/18

Bipartite Transition System: A New Approach

 Bipartite transition system

What is our new approach?

  • A game structure between the controller and the system
  • Enumerates all (infinite) legal solutions using a finite structure
  • A state-based approach for synthesis
  • Inspired by methodologies in reactive synthesis literature

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

slide-13
SLIDE 13

9/18

Bipartite Transition System

Information State: a set of states, 𝐽 ≔ 2π‘Œ

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

slide-14
SLIDE 14

9/18

Bipartite Transition System

  • Definition. (BTS).

A bipartite transition system T w.r.t. G is a 7-tuple π‘ˆ = (𝑅𝑍, π‘…π‘Ž, β„Žπ‘π‘Ž, β„Žπ‘Žπ‘, 𝐹, Ξ“, 𝑧0) where

  • 𝑅𝑍 βŠ† 𝐽 is the set of Y-states;
  • π‘…π‘Ž βŠ† 𝐽 Γ— Ξ“ is the set of Z-states so that z = (𝐽 𝑨 , Ξ“ 𝑨 );
  • β„Žπ‘π‘Ž: 𝑅𝑍 Γ— Ξ“ β†’ Qπ‘Ž represents the unobservable reach;
  • β„Žπ‘Žπ‘: π‘…π‘Ž

π‘ˆ Γ— E β†’ Q𝑍 represents the observation transition;

  • E is the set of events of G;
  • Ξ“ is the set of admissible control decisions of G;
  • 𝑧0 = {𝑦0} is the initial state.

Information State: a set of states, 𝐽 ≔ 2π‘Œ

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

slide-15
SLIDE 15

10/18

Bipartite Transition System

1 3 5 7 11 12 15 8 2 6 4 9 10 14 13 𝑐1 𝑐2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑝2 𝑝1

{0}

𝐹𝑑 = {𝑑1, 𝑑2}, 𝐹𝑝 = {𝑝1, 𝑝2}

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

Illegal states= π‘Œ βˆ– π‘ŒπΌ = {15}

slide-16
SLIDE 16

10/18

Bipartite Transition System

1 3 5 7 11 12 15 8 2 6 4 9 10 14 13 𝑐1 𝑐2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑝2 𝑝1

{0}

{ }

𝐹𝑑 = {𝑑1, 𝑑2}, 𝐹𝑝 = {𝑝1, 𝑝2}

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

Illegal states= π‘Œ βˆ– π‘ŒπΌ = {15}

slide-17
SLIDE 17

10/18

Bipartite Transition System

1 3 5 7 11 12 15 8 2 6 4 9 10 14 13 𝑐1 𝑐2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑝2 𝑝1

{0,1,2},{ } {0}

{ }

𝐹𝑑 = {𝑑1, 𝑑2}, 𝐹𝑝 = {𝑝1, 𝑝2}

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

Illegal states= π‘Œ βˆ– π‘ŒπΌ = {15}

slide-18
SLIDE 18

10/18

Bipartite Transition System

1 3 5 7 11 12 15 8 2 6 4 9 10 14 13 𝑐1 𝑐2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑝2 𝑝1

{0,1,2},{ } {0} 𝑝1

{ }

𝐹𝑑 = {𝑑1, 𝑑2}, 𝐹𝑝 = {𝑝1, 𝑝2}

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

Illegal states= π‘Œ βˆ– π‘ŒπΌ = {15}

slide-19
SLIDE 19

10/18

Bipartite Transition System

1 3 5 7 11 12 15 8 2 6 4 9 10 14 13 𝑐1 𝑐2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑝2 𝑝1

{0,1,2},{ } {0}

{ }

𝐹𝑑 = {𝑑1, 𝑑2}, 𝐹𝑝 = {𝑝1, 𝑝2}

{3,4} 𝑝1 X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

Illegal states= π‘Œ βˆ– π‘ŒπΌ = {15}

slide-20
SLIDE 20

10/18

Bipartite Transition System

1 3 5 7 11 12 15 8 2 6 4 9 10 14 13 𝑐1 𝑐2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑝2 𝑝1

{0,1,2},{ } {0} {3,4} 𝑝1

{ } {𝑑1}

𝐹𝑑 = {𝑑1, 𝑑2}, 𝐹𝑝 = {𝑝1, 𝑝2}

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

Illegal states= π‘Œ βˆ– π‘ŒπΌ = {15}

slide-21
SLIDE 21

10/18

Bipartite Transition System

1 3 5 7 11 12 15 8 2 6 4 9 10 14 13 𝑐1 𝑐2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑝2 𝑝1

{0,1,2},{ } {0} {3,4} {3,4,7,10} {𝑑1} 𝑝1

{ } {𝑑1}

𝐹𝑑 = {𝑑1, 𝑑2}, 𝐹𝑝 = {𝑝1, 𝑝2}

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

Illegal states= π‘Œ βˆ– π‘ŒπΌ = {15}

slide-22
SLIDE 22

10/18

Bipartite Transition System

1 3 5 7 11 12 15 8 2 6 4 9 10 14 13 𝑐1 𝑐2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑝2 𝑝1

{0,1,2},{ } {0} {3,4} {3,4,7,10} {𝑑1} 𝑝1 𝑝1

{ } {𝑑1}

𝐹𝑑 = {𝑑1, 𝑑2}, 𝐹𝑝 = {𝑝1, 𝑝2}

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

Illegal states= π‘Œ βˆ– π‘ŒπΌ = {15}

slide-23
SLIDE 23

10/18

Bipartite Transition System

1 3 5 7 11 12 15 8 2 6 4 9 10 14 13 𝑐1 𝑐2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑝2 𝑝1

{0,1,2},{ } {0} {3,4} {3,4,7,10} {𝑑1} 𝑝1 𝑝1

{ } {𝑑1}

𝐹𝑑 = {𝑑1, 𝑑2}, 𝐹𝑝 = {𝑝1, 𝑝2}

{1,2} X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

Illegal states= π‘Œ βˆ– π‘ŒπΌ = {15}

slide-24
SLIDE 24

11/18

Total Controller

Total Controller A BTS π‘ˆ that enumerates all control decisions at Y and all observations at Z

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

slide-25
SLIDE 25

11/18

Total Controller

1 3 5 7 11 12 15 8 2 6 4 9 10 14 13 𝑐1 𝑐2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑝2 𝑝1 {0,1,2},{ } {0} {3,4} {5,6} {3,4,7,10} {𝑑1} {3,4,8,9} {𝑑2} {5,6,12,13} {𝑑1} {5,6,11,14} {𝑑2} {1,2} {1,2 },{ } {3,4},{ } {5,6},{ } 𝑝1 𝑝2 𝑝1 𝑝2

{ } { } { } { } {𝑑1} {𝑑2} {𝑑1} {𝑑2}

𝑝1 𝑝2

𝐹𝑑 = {𝑑1, 𝑑2}, 𝐹𝑝 = {𝑝1, 𝑝2}

{𝑑1, 𝑑2} {3,4,7,8,9, 10,15} ,{𝑑1, 𝑑2} {5,6,11,12,13, 14,15} ,{𝑑1, 𝑑2} {𝑑1, 𝑑2}

Total Controller A BTS π‘ˆ that enumerates all control decisions at Y and all observations at Z

𝑝1 𝑝2 X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

Illegal states= π‘Œ βˆ– π‘ŒπΌ = {15}

slide-26
SLIDE 26

12/18

Safety Binary Function

𝐸𝐽 𝑗 = 1, 0, if βˆ€π‘¦ ∈ 𝑗: 𝑦 ∈ π‘ŒπΌ

  • therwise

Safety Binary function for Information State: 𝐸𝐽: 𝐽 β†’ {0,1}

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

slide-27
SLIDE 27

12/18

Safety Binary Function

𝐸𝑍 𝑧 = 1, 0, πΈπ‘Ž(𝑨) = 1, 0, if 𝐸𝐽 𝑧 = 1 π‘π‘œπ‘’ βˆƒπ›Ώ ∈ Ξ“: πΈπ‘Ž β„Žπ‘π‘Ž 𝑧, 𝛿 = 1

  • therwise

if 𝐸𝐽 𝐽(𝑨) = 1 π‘π‘œπ‘’ βˆ€π‘“ ∈ 𝛿 ∩ 𝐹𝑝: 𝐸𝑍 β„Žπ‘Žπ‘ 𝑨, 𝑓 = 1

  • therwise

Safety Binary function for Y and Z-states: 𝐸𝑍: 𝐽 β†’ {0,1} and πΈπ‘Ž: 𝐽 Γ— Ξ“ β†’ {0,1} 𝐸𝐽 𝑗 = 1, 0, if βˆ€π‘¦ ∈ 𝑗: 𝑦 ∈ π‘ŒπΌ

  • therwise

Safety Binary function for Information State: 𝐸𝐽: 𝐽 β†’ {0,1}

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

slide-28
SLIDE 28

13/18

All Inclusive Controller

The All Inclusive Controller for 𝐻 π’β„π’Ÿ 𝐻 = (𝑅𝑍

𝐡𝐽𝐷𝐻, π‘…π‘Ž 𝐡𝐽𝐷𝐻, β„Žπ‘π‘Ž 𝐡𝐽𝐷𝐻, β„Žπ‘Žπ‘ 𝐡𝐽𝐷𝐻, 𝐹, Ξ“, 𝑧_0),

is defined as the largest BTS consisting of only safe reachable Y and Z-states, and the transitions between them. All Inclusive Controller:

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

slide-29
SLIDE 29

13/18

All Inclusive Controller

The All Inclusive Controller for 𝐻 π’β„π’Ÿ 𝐻 = (𝑅𝑍

𝐡𝐽𝐷𝐻, π‘…π‘Ž 𝐡𝐽𝐷𝐻, β„Žπ‘π‘Ž 𝐡𝐽𝐷𝐻, β„Žπ‘Žπ‘ 𝐡𝐽𝐷𝐻, 𝐹, Ξ“, 𝑧_0),

is defined as the largest BTS consisting of only safe reachable Y and Z-states, and the transitions between them. All Inclusive Controller:

1 3 5 7 11 12 15 8 2 6 4 9 10 14 13 𝑐1 𝑐2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑝2 𝑝1 𝐹𝑑 = {𝑑1, 𝑑2}, 𝐹𝑝 = {𝑝1, 𝑝2}

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014 {0,1,2},{ } {0} {3,4} {5,6} {3,4,7,10} {𝑑1} {3,4,8,9} {𝑑2} {5,6,12,13} {𝑑1} {5,6,11,14} {𝑑2} {1,2} {1,2 },{ } {3,4},{ } {5,6},{ } 𝑝1 𝑝2 𝑝1 𝑝2

{ } { } { } { } {𝑑1} {𝑑2} {𝑑1} {𝑑2}

𝑝1 𝑝2

{𝑑1, 𝑑2} {3,4,7,8,9, 10,15} ,{𝑑1, 𝑑2} {5,6,11,12,13, 14,15} ,{𝑑1, 𝑑2} {𝑑1, 𝑑2}

𝑝1 𝑝2

slide-30
SLIDE 30

13/18

All Inclusive Controller

The All Inclusive Controller for 𝐻 π’β„π’Ÿ 𝐻 = (𝑅𝑍

𝐡𝐽𝐷𝐻, π‘…π‘Ž 𝐡𝐽𝐷𝐻, β„Žπ‘π‘Ž 𝐡𝐽𝐷𝐻, β„Žπ‘Žπ‘ 𝐡𝐽𝐷𝐻, 𝐹, Ξ“, 𝑧_0),

is defined as the largest BTS consisting of only safe reachable Y and Z-states, and the transitions between them. All Inclusive Controller:

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

1 3 5 7 11 12 15 8 2 6 4 9 10 14 13 𝑐1 𝑐2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑝1 𝑝2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑑1 𝑑2 𝑝2 𝑝1 𝐹𝑑 = {𝑑1, 𝑑2}, 𝐹𝑝 = {𝑝1, 𝑝2}

{0,1,2},{ } {0} {3,4} {5,6} {3,4,7,10} {𝑑1} {3,4,8,9} {𝑑2} {5,6,12,13} {𝑑1} {5,6,11,14} {𝑑2} {1,2} {1,2 },{ } {3,4},{ } {5,6},{ } 𝑝1 𝑝2 𝑝1 𝑝2

{ } { } { } { } {𝑑1} {𝑑2} {𝑑1} {𝑑2}

𝑝1 𝑝2

{𝑑1, 𝑑2} {3,4,7,8,9, 10,15} ,{𝑑1, 𝑑2} {5,6,11,12,13, 14,15} ,{𝑑1, 𝑑2} {𝑑1, 𝑑2}

𝑝1 𝑝2

Illegal states= π‘Œ βˆ– π‘ŒπΌ = {15}

slide-31
SLIDE 31

14/18

Construction of the AIC

 Pruning states from the total controller 𝑃 (2 π‘Œ

2)

Construction of the AIC

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

slide-32
SLIDE 32

14/18

Construction of the AIC

 Pruning states from the total controller 𝑃 (2 π‘Œ

2)

Construction of the AIC

  • Pre-compute the extended specification
  • π‘Š 𝑦 = ∞ ⇔ βˆƒπ‘‘ ∈ 𝐹𝑣𝑑

βˆ— : 𝑔 𝑦, 𝑑 ∈ π‘Œ βˆ– π‘ŒπΌ

  • 𝐸𝑍 𝑧 = 0 ⇔ βˆƒπ‘¦ ∈ 𝑧: π‘Š 𝑦 = ∞
  • Construct the AIC by a DFS

 Our Approach 𝑃(2|π‘Œ|)

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

slide-33
SLIDE 33

15/18

Properties of the AIC

Theorem. ( 𝑀 = 𝑀 βŠ† β„’ 𝐼 ∧ 𝑀 is observable ∧ 𝑀 is controllable) ⇔ 𝑀 ∈ β„’π‘ˆπ‘‡(π’β„π’Ÿ(𝐻)) Theorem. There exists a safe partial observation supervisor if and only if π’β„π’Ÿ(𝐻) is non-empty.

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

slide-34
SLIDE 34

15/18

Properties of the AIC

Theorem. ( 𝑀 = 𝑀 βŠ† β„’ 𝐼 ∧ 𝑀 is observable ∧ 𝑀 is controllable) ⇔ 𝑀 ∈ β„’π‘ˆπ‘‡(π’β„π’Ÿ(𝐻)) Theorem. There exists a safe partial observation supervisor if and only if π’β„π’Ÿ(𝐻) is non-empty.

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

{0,1,2},{ } {0} {3,4} {5,6} {3,4,7,10} {𝑑1} {3,4,8,9} {𝑑2} {5,6,12,13} {𝑑1} {5,6,11,14} {𝑑2} {1,2} {1,2 },{ } {3,4},{ } {5,6},{ } 𝑝1 𝑝2 𝑝1 𝑝2 { } { } { } { } {𝑑1} {𝑑2} {𝑑1} {𝑑2} 𝑝1 𝑝2

slide-35
SLIDE 35

15/18

Properties of the AIC

Theorem. ( 𝑀 = 𝑀 βŠ† β„’ 𝐼 ∧ 𝑀 is observable ∧ 𝑀 is controllable) ⇔ 𝑀 ∈ β„’π‘ˆπ‘‡(π’β„π’Ÿ(𝐻)) Theorem. There exists a safe partial observation supervisor if and only if π’β„π’Ÿ(𝐻) is non-empty.

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

{0,1,2},{ } {0} {3,4} {5,6} {3,4,7,10} {𝑑1} {3,4,8,9} {𝑑2} {5,6,12,13} {𝑑1} {5,6,11,14} {𝑑2} {1,2} {1,2 },{ } {3,4},{ } {5,6},{ } 𝑝1 𝑝2 𝑝1 𝑝2 { } { } { } { } {𝑑1} {𝑑2} {𝑑1} {𝑑2} 𝑝1 𝑝2 {0,1,2},{ } {0} {3,4} {5,6} {3,4,7,10} {𝑑1} {3,4,8,9} {𝑑2} {5,6,12,13} {𝑑1} {5,6,11,14} {𝑑2} {1,2} {1,2 },{ } {3,4},{ } {5,6},{ } 𝑝1 𝑝2 𝑝1 𝑝2 { } { } { } { } {𝑑1} {𝑑2} {𝑑1} {𝑑2} 𝑝1 𝑝2

slide-36
SLIDE 36

15/18

Properties of the AIC The AIC embeds all legal solutions!

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

Theorem. ( 𝑀 = 𝑀 βŠ† β„’ 𝐼 ∧ 𝑀 is observable ∧ 𝑀 is controllable) ⇔ 𝑀 ∈ β„’π‘ˆπ‘‡(π’β„π’Ÿ(𝐻)) Theorem. There exists a safe partial observation supervisor if and only if π’β„π’Ÿ(𝐻) is non-empty.

slide-37
SLIDE 37

16/18

State Based Property of the AIC

  • Definition. (IS-Based Supervisor)

A partial observation supervisor 𝑇𝑄 is said to be information-state-based if βˆ€π‘‘, 𝑒 ∈ β„’ 𝑇𝑄 𝐻 [𝐽𝑇𝑇𝑄

𝑍 𝑑 = 𝐽𝑇𝑇𝑄 𝑍 𝑒 β‡’ 𝑇𝑄 𝑑 = 𝑇𝑄(𝑒)]

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

slide-38
SLIDE 38

16/18

State Based Property of the AIC

  • Definition. (IS-Based Supervisor)

A partial observation supervisor 𝑇𝑄 is said to be information-state-based if βˆ€π‘‘, 𝑒 ∈ β„’ 𝑇𝑄 𝐻 [𝐽𝑇𝑇𝑄

𝑍 𝑑 = 𝐽𝑇𝑇𝑄 𝑍 𝑒 β‡’ 𝑇𝑄 𝑑 = 𝑇𝑄(𝑒)]

Theorem.

There exists at least one IS-based supervisor 𝑇𝐽 such that β„’ 𝑇𝐽 𝐻 is a maximal safe, controllable and observable sublanguage.

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

slide-39
SLIDE 39

16/18

State Based Property of the AIC

  • Definition. (IS-Based Supervisor)

A partial observation supervisor 𝑇𝑄 is said to be information-state-based if βˆ€π‘‘, 𝑒 ∈ β„’ 𝑇𝑄 𝐻 [𝐽𝑇𝑇𝑄

𝑍 𝑑 = 𝐽𝑇𝑇𝑄 𝑍 𝑒 β‡’ 𝑇𝑄 𝑑 = 𝑇𝑄(𝑒)]

Theorem.

There exists at least one IS-based supervisor 𝑇𝐽 such that β„’ 𝑇𝐽 𝐻 is a maximal safe, controllable and observable sublanguage.

Our information state is correctly defined for safety specification

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

slide-40
SLIDE 40

17/18

Synthesis of Safe and Maximally Permissive Supervisors

{0,1,2},{ } {0} {3,4} {5,6} {3,4,7,10} {𝑑1} {3,4,8,9} {𝑑2} {5,6,12,13} {𝑑1} {5,6,11,14} {𝑑2} {1,2} {1,2 },{ } {3,4},{ } {5,6},{ } 𝑝1 𝑝2 𝑝1 𝑝2 { } { } { } { } {𝑑1} {𝑑2} {𝑑1} {𝑑2} 𝑝1 𝑝2

Synthesis Step:

  • 1. Build the AIC
  • 2. For any Y-state, pick one local maximal control

decision

  • 3. For any Z-state, pick all observations
  • 4. Until reach a terminal state or a state that

has been visited

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

slide-41
SLIDE 41

17/18

Synthesis of Safe and Maximally Permissive Supervisors

{0,1,2},{ } {0} {3,4} {5,6} {3,4,7,10} {𝑑1} {3,4,8,9} {𝑑2} {5,6,12,13} {𝑑1} {5,6,11,14} {𝑑2} {1,2} {1,2 },{ } {3,4},{ } {5,6},{ } 𝑝1 𝑝2 𝑝1 𝑝2 { } { } { } { } {𝑑1} {𝑑2} {𝑑1} {𝑑2} 𝑝1 𝑝2

Synthesis Step:

  • 1. Build the AIC
  • 2. For any Y-state, pick one local maximal control

decision

  • 3. For any Z-state, pick all observations
  • 4. Until reach a terminal state or a state that

has been visited

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

slide-42
SLIDE 42

17/18

Synthesis of Safe and Maximally Permissive Supervisors

{0,1,2},{ } {0} {3,4} {5,6} {3,4,7,10} {𝑑1} {3,4,8,9} {𝑑2} {5,6,12,13} {𝑑1} {5,6,11,14} {𝑑2} {1,2} {1,2 },{ } {3,4},{ } {5,6},{ } 𝑝1 𝑝2 𝑝1 𝑝2 { } { } { } { } {𝑑1} {𝑑2} {𝑑1} {𝑑2} 𝑝1 𝑝2

Synthesis Step:

  • 1. Build the AIC
  • 2. For any Y-state, pick one local maximal control

decision

  • 3. For any Z-state, pick all observations
  • 4. Until reach a terminal state or a state that

has been visited

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

slide-43
SLIDE 43

17/18

Synthesis of Safe and Maximally Permissive Supervisors

{0,1,2},{ } {0} {3,4} {5,6} {3,4,7,10} {𝑑1} {3,4,8,9} {𝑑2} {5,6,12,13} {𝑑1} {5,6,11,14} {𝑑2} {1,2} {1,2 },{ } {3,4},{ } {5,6},{ } 𝑝1 𝑝2 𝑝1 𝑝2 { } { } { } { } {𝑑1} {𝑑2} {𝑑1} {𝑑2} 𝑝1 𝑝2

Synthesis Step:

  • 1. Build the AIC
  • 2. For any Y-state, pick one local maximal control

decision

  • 3. For any Z-state, pick all observations
  • 4. Until reach a terminal state or a state that

has been visited

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

slide-44
SLIDE 44

17/18

Synthesis of Safe and Maximally Permissive Supervisors

{0,1,2},{ } {0} {3,4} {5,6} {3,4,7,10} {𝑑1} {3,4,8,9} {𝑑2} {5,6,12,13} {𝑑1} {5,6,11,14} {𝑑2} {1,2} {1,2 },{ } {3,4},{ } {5,6},{ } 𝑝1 𝑝2 𝑝1 𝑝2 { } { } { } { } {𝑑1} {𝑑2} {𝑑1} {𝑑2} 𝑝1 𝑝2

Synthesis Step:

  • 1. Build the AIC
  • 2. For any Y-state, pick one local maximal control

decision

  • 3. For any Z-state, pick all observations
  • 4. Until reach a terminal state or a state that

has been visited

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

slide-45
SLIDE 45

17/18

Synthesis of Safe and Maximally Permissive Supervisors

{0,1,2},{ } {0} {3,4} {5,6} {3,4,7,10} {𝑑1} {3,4,8,9} {𝑑2} {5,6,12,13} {𝑑1} {5,6,11,14} {𝑑2} {1,2} {1,2 },{ } {3,4},{ } {5,6},{ } 𝑝1 𝑝2 𝑝1 𝑝2 { } { } { } { } {𝑑1} {𝑑2} {𝑑1} {𝑑2} 𝑝1 𝑝2

Synthesis Step:

  • 1. Build the AIC
  • 2. For any Y-state, pick one local maximal control

decision

  • 3. For any Z-state, pick all observations
  • 4. Until reach a terminal state or a state that

has been visited

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

slide-46
SLIDE 46

18/18

Summary

Contribution:

  • A new bipartite transition system that captures all safe decisions

in a single finite structure.

  • Construction of the AIC
  • Properties of the AIC
  • Synthesis of supervisors based on the AIC

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014

slide-47
SLIDE 47

18/18

Summary

Contribution:

  • A new bipartite transition system that captures all safe decisions

in a single finite structure.

  • Construction of the AIC
  • Properties of the AIC
  • Synthesis of supervisors based on the AIC

Future Work:

  • Optimal synthesis problem
  • Non-blocking specification
  • Decentralized synthesis problem

X.Yin & S.Lafortune (UMich) August 25, 2014 IFAC World Congress 2014