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Semi-asynchronous Fault Diagnosis of Discrete Event Systems

ALEJANDRO WHITE

• DR. ALI KARIMODDINI

NC A&T State University

http://accesslab.net/ http://techlav.ncat.edu/ http://www.ncat.edu/

OCTOBER 2017

Table Of Contents

Motivation Background Problem Statement System Under Diagnosis Diagnoser Semi-Asynchronous Diagnosability Simulated Example Conclusion

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 2

Why Do We Need Fault Diagnosis?

As systems have become larger, more complex, and more integrated into our daily lives, it is imperative and obligatory that there exists systematic fault diagnosis techniques that provide a timely and accurate diagnosis of system behaviors

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 3

Crash Report

Perrone Robotics driverless car crashes after being hacked during testing.

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 4

Increase of roles and presence in daily societal activities leads to an increase in liability

American Airlines Flight 191 (1979)

• Left Engine separated from wing
• Pilot only 15s to react
• Subsequent analysis shows consequence of faults avoidable

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 5

Failure to properly diagnose faults, leads to incorrect recovery actions

General Objectives & Impacts of Fault Diagnosis

Impacts

Upon a fault

• ccurrence,

a system will autonomously become aware

• f

the fault’s

• ccurrence, and initiate a systematic procedure that

locates, identifies, and accommodates the fault in

• rder to ensure proper utilization of the system’s

remaining resources, allowing for a resilient post fault system operation that is both safe and stable.

Objectives

To develop systematic techniques for automatic diagnosis of faults in the system to timely diagnose (detect, identify and locate) occurred system faults. Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 6

Why Discrete Event System (DES)?

Topology The topology of a DES, represents a system’s behavior as sequences of discrete

• events. This allows for the

capturing of disruptive changes in a system’s

• peration; in turn highlighting

faulty behaviors of the system.

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 7

Preliminaries and Background

Automaton

 State space (𝒀): a discrete set of system states  Event set (𝜯 = 𝜯𝒑⋃𝜯𝒗): notable occurrences of asynchronous discrete changes in a system  Observable events (𝜯𝒑): events observed by a sensor (e.g., flowing of water)  Unobservable events (𝜯𝒗): events that are unable to be detected by sensors; possibly due to sensor absence/damage (e.g., failure event)  State-transition relation (𝜺: 𝒀 × 𝜯 → 𝟑𝒀): a partial relation that determines all feasible system state transitions caused by system events (𝟑𝒀 is the set of all possible combinations of states)  Initial state (𝒚𝟏): indicated by an input arrow connected to a single state

 Definition: a non-deterministic finite-state Discrete-Event System (DES) can be represented by a four-tuple: 𝐇 = (𝐘, 𝚻, 𝛆, 𝒚𝟏)

𝛽 𝑣 𝛽 𝛾

 𝒀 = {𝟐, 𝟑, 𝟒}  𝚻 = {𝐯, 𝜷, 𝜸}  𝚻𝒑 = {𝜷, 𝜸}  𝚻𝒗 = {𝒗}  𝜺 𝟐, 𝒗 = 𝟑, 𝜺 𝟑, 𝜷 = {𝟑, 𝟒}, 𝜺 𝟒, 𝜸 = 𝟒  𝒚𝟏 = {𝟐} Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 9

 Trace (string): a sequence of one or more events, allowable by the system’s behavior  e.g., 𝒕 = 𝒇𝟐𝒇𝟑 … 𝒇𝒐 where 𝒇𝒋 ∈ 𝜯  Language (𝓜𝑯(𝒚𝟏)): the set of all system traces which originate at the system’s initial state 𝒚𝟏  𝓜𝑯(𝒚𝟏) = {𝒕 ∈ 𝚻∗|𝜺 𝒚𝟏, 𝒕 𝐣𝐭 𝐞𝐟𝐠𝐣𝐨𝐟𝐞} (𝚻∗ is the Kleene Closure of 𝚻)

Automaton (Language)

 Definition: the system language is a discrete representation of the system’s behaviors (normal and faulty) in the form of sequences of events

𝛽 𝑣 𝛽 𝛾

 ℒ𝑯 𝒚𝟏 = {𝜻, 𝒗, 𝒗𝜷∗, 𝒗𝜷∗𝜷, 𝒗𝜷∗𝜸∗, 𝒗𝜷∗𝜸𝜸∗, … } = {𝜻, 𝒗𝜷∗𝜸∗} Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 10 *: arbitrarily repeated string

 Our purpose is to diagnose unobservable faults from the observable behavior of the system.  The system’s observable behavior can be described by the natural projection (𝑄) of the system’s language to the observable language set of the system.

Natural Projection

𝛽 𝑣 𝛽 𝛾

 ℒ𝑯 𝒚𝟏 = {𝒗, 𝒗𝜷∗𝜸∗}  𝐐(ℒ𝑯 𝒚𝟏 ) = {𝜻, 𝜷∗𝜸∗}  𝑄

ℒ𝑯(𝒚𝟏) −1

𝜷 = {𝒗𝜷}

𝑸: 𝚻∗ → 𝚻𝒑

∗

𝑸 𝜻 = 𝜻 𝑸 𝒇 = 𝒇 if 𝒇 ∈ 𝚻𝟏 𝑸 𝒇 = 𝜻 if 𝒇 ∉ 𝚻𝒑 𝑸(𝒕𝒇) = 𝑸 𝒕 𝑸 𝒇 for 𝒕 ∈ 𝚻∗ and 𝒇 ∈ 𝚻 Extension of the natural projection to the languages: 𝑄 ℒ𝐇(𝐲𝟏) = {𝑄 𝑡 ∣ 𝑡 ∈ ℒ𝐇(𝐲𝟏)} Inverse of Natural Projection 𝑄

ℒ𝑯(𝒚𝟏) −1

𝑥 = {𝑡 ∈ ℒ𝐡(𝐲𝟏) ∣ 𝑄 𝑡 = 𝑥}

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 11

 Here we present pre-defined sets of system strings

System String Sets

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 12  Extension Closure: 𝐟𝐲𝐮 ℒ𝑯 𝒚𝟏 ≔ {𝒘 ∈ 𝚻∗ ∣ ∃𝒗 ∈ ℒ𝑯 𝒚𝟏 : 𝒗𝒘 ∈ ℒ𝑯(𝒚𝟏)}  𝑸𝒔𝒇 𝒀𝒕 : 𝑸𝒔𝒇 𝒀𝒕 = {𝒕 ∈ ℒ𝑯 𝒚𝟏 ∣ 𝜺 𝒚𝟏, 𝒕 ∈ 𝒀𝒕}, the set of strings leading to 𝒀𝒕 generated from 𝒚𝟏  𝑸𝒑𝒕𝒖 𝒀𝒕 : the set of strings that can be generated from any 𝒚 ∈ 𝒀𝒕  Unobservable Reach: 𝑽𝑺 𝒕 = {𝒛 ∈ 𝒀 ∣ ∃𝒗 ∈ 𝚻𝐯

∗, 𝜺 𝒚, 𝒗 = 𝒛}, the set of all system states (with the inclusion of x itself)

reachable from state 𝒚 via strings solely consisting of unobservable events  Unobservable Extension: 𝑽𝑭 𝒕, 𝒚 = {𝒕. 𝒖 ∣ 𝒖 ∈ 𝚻𝒗

∗ 𝒃𝒐𝒆 𝒕. 𝒖 ∈ ℒ𝑯(𝒚)}, specifies the set of all unobservable extensions of 𝒕

concatenated with the string 𝒕, and generated from the state x

Diagnoser Functionality

Fault Detection Fault Isolation Fault Identification

 Fault diagnostics is provided by the diagnoser.  The diagnoser extracts information from the original system’s observable behaviors, in order to estimate the original system’s current state and current condition (faulty or non-faulty).  The diagnoser’s transitions are only defined over the original system’s observable event

• ccurrences.

 Upon observance of the original system’s behavior, the diagnoser updates its estimation of the

• riginal system’s state and condition.

𝓜𝑯(𝒚𝟏) 𝑸(𝓜𝑯(𝒚𝟏)) Natural Projection 𝑸: 𝚻∗ → 𝚻𝒑

∗

Plant 𝑯 = (𝒀, 𝚻, 𝜺, 𝒚𝟏) Diagnoser

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 13

Fault Diagnosis

Centralized: Sampath et al. 1995, Zad et al. 2003 Decentralized: Wang et al. 2007, Lafortune et al. 2005 Distributed: Fabre 2002, Pencolé 2005 Event-based: Sampath et al.1995 State-based: Lin 1994, Zad et

• al. 2003

Petri Nets: Prock 1991, Basile et al. 2008 Process Algebra: Console et al. 2002 Automata Theory: Sampath 1995, Wang et al., 2007

DES Fault Diagnosis (State-Of-The-Art)

TOOLS STRUCTURE STRATEGY Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 14

Bridging Gaps In DES Diagnosis Methods

In many practical situations,

• nly after a fault occurs, the

diagnosis tool can be brought in and connected to the faulty plant to diagnose the occurred fault. In many cases, it is not possible, or it is time-consuming and costly, to restart the plant to be synchronized with the diagnoser.

In all of existing methods, the diagnoser has to be simultaneously initialized with the system under diagnosis.

In all of existing methods, the diagnoser should synchronously execute the events in parallel with system under diagnosis, to keep the past history of exhibited normal and faulty behaviors. Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 15

Objectives of This Research

To develop automatic diagnosis techniques to timely diagnose (detect, identify and locate) occurred faults. To develop a diagnosis approach that can definitively diagnose all modelled system fault occurrences. To construct a semi-asynchronous DES fault diagnoser, which is not required to be synchronously initialized with the system under diagnosis (i.e., it can work without requiring the restarting of the system).

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 16

Unlike conventional diagnosis techniques, the past history of the system before the activation of the semi-asynchronous diagnoser is not available, leaving the semi-asynchronous diagnoser with the challenge of diagnosing faults using only the future behaviors of the plant observed after the activation

• f the semi-asynchronous diagnoser.

In contrast to existing methods, where the initial state of the system and correspondingly the initial state of the diagnoser are generally assumed to be non-faulty; upon its initialization, the semi-asynchronous diagnoser is not able to assume that the current condition of the system is normal.

Challenges

I. II.

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 17

Model-based, and definition- based verification of 𝑮𝒋-Semi- Asynchronous Diagnosability Introduced formal definition of 𝑮𝒋-Semi- Asynchronous Diagnosability Novel diagnoser capable of diagnosing multiple-typed system faults without the commonplace requirement of system reinitialization

Contributions

DES Fault Diagnosis

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 18

Problem Statement (Semi-Asynchronous Diagnosis)

When there is uncertain information about the state of the system upon activation of the diagnoser, how do we distinctively characterize a system’s behavior (system state and condition) solely based upon a finite number of subsequent successive external system

• bservations.

Observations Diagnosis

Fault

HYPOTHESIZED SYSTEM STATE CONDITION AND LOCATIONS

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 19

Proposed Diagnosis Approach

Problem Formulation

Consider the DES plant 𝐻 = 𝑌, Σ, 𝜀, 𝑦0 where 𝑌, Σ, 𝜀 and 𝑦0 represent the system’s state space, event set, state transition relation, and the system’s initial state, respectively. Consider that the event set Σ = Σ𝑝 ∪ Σ𝑣 can be disjointly partitioned into two subsets: Σ𝑝 (the observable event set) and Σ𝑣 (the unobservable event set which consists of unobservable events including faults Σ𝑔 ⊆ Σ𝑣). Consider 𝑌𝑡 as the initial estimation of possible state locations of plant 𝐻, upon the activation of the diagnoser.

Then

• For any string 𝒕 that belongs to

the language of the plant (𝒕 ∈ ℒ𝑯(𝒚𝟏)) that leads to one of the states in 𝒀𝒕 (𝜺 𝒚𝟏, 𝒕 ∈ 𝒀𝒕).

• And for any of its sufficiently

long suffix strings 𝒖 (𝒖 ∈ ℒ𝑯/𝒕) where 𝒖 occurs after the diagnoser activation.

Determine from the

• bservable behavior of the

system, 𝑸 𝒖

• If a fault has occurred (i.e., check

if ∃𝒈 ∈ 𝚻𝒈 𝒕𝒗𝒅𝒊 𝒖𝒊𝒃𝒖 𝒈 ∈ 𝒕. 𝒖)

• If yes, determine the type of the

fault 𝚻𝒈𝒋 where 𝒈 ∈ 𝚻𝒈𝒋

• Locate the system state 𝒚 ∈ 𝒀

that is subsequently reached by 𝒕. 𝒖 Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 20

Proposed Solution

We propose a novel algorithm that generates a deterministic finite state event-based diagnoser capable of diagnosing multiple, non-permanent system faults without requiring the restarting of the system under diagnosis.

The algorithm produced semi-asynchronous diagnoser, starts with an uncertain estimation of the system under diagnosis, and upon gathering run-time system observations, the diagnoser refines its estimates of the original system’s state and condition Diagnoser capable of diagnosing system faults the occur before and after diagnoser activation Our design takes advantage of the fact that many systems exhibit behaviors where the system’s possible state locations can be deduced to a set of states, or that a system can be derived to a situation where its possible system state locations can be deduced to a set of states.

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 21

• A. White, A. Karimoddini, “Semi-Asynchronous Fault Diagnosis of Discrete Event Systems,” The 2016 IEEE International Conference on Systems, Man, and Cybernetics.

Assumptions

FAULTS ARE UNOBSERVABLE (𝚻𝒈 ⊆ 𝚻𝒗 ⊆ 𝚻) Otherwise their detection would be trivial MODELLED FAULTS DO NOT BRING THE SYSTEM TO A HALT Providing enough time to diagnose the fault before the system crashes NO ARBITRARILY LONG STRINGS OF UNOBSERVABLE EVENTS (∀𝒕𝒗𝒘 ∈ 𝓜𝑯, 𝒕, 𝒘 ∈ 𝚻𝟏

∗, 𝒗 ∈ 𝚻𝒗 ∗, ∃𝒐 ∈ ℕ, such that 𝒗

≤ 𝒐) This ensures that following the occurrence of an unobservable event, sooner or later the system will produce an observable event. Otherwise the system may get stuck in an infinitely long unobservable string of events, which prevents any diagnosis about what is going on inside the system LIVE LANGUAGE (∀𝒚 ∈ 𝒀, ∃𝒇 ∈ 𝚻 such that 𝜺 𝒚, 𝒇 is defined) This is to ensure that in the future the system will always produce a sufficiently long string of

• bservable events to be used for diagnosis.

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 22

The developed algorithm produces diagnoser states 𝑟 ∈ 𝑅𝑒; each diagnoser state 𝑟 is a set of

• rdered pairs composed of the diagnoser’s estimates of system state 𝑦, and system condition ℓ.

System Condition Labels 𝑴 𝐎 : Normal Operation 𝐆 = {𝐆𝟐, 𝐆𝟑, … , 𝐆𝐧}: Fault labels for fault type events 𝚻𝐠𝐣, 𝐣 = 𝟐, … , 𝐧 Diagnoser State Composition: 𝐫 = 𝐲𝟐, ℓ𝟐 , … , 𝐲𝐥, ℓ𝐥 𝐲𝐤 ∈ 𝐘, ℓ𝐤∈ 𝐌 = 𝐎 ⋃𝟑𝐆 𝐤 = 𝟐, … , 𝐥

Functionality of Algorithm Offline Generate entire diagnoser Online Provide on-line diagnostics

Diagnoser state 𝒓 is considered an 𝑮𝒋-certain state if: 𝑮𝒋 ∈ ℓ𝒌 ∀𝒌

The Developed Algorithm for Diagnoser Construction

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 23

The Developed Algorithm for Diagnoser Construction

Step 1: Constructing 𝒓𝟏 i. Since it is assumed that the initial state of the original system 𝑯 is normal (non-faulty), the algorithm begins by setting 𝒓𝒕 = 𝒚𝟏, 𝑶 . ii. Next, 𝒓𝒕 is extended to include its unobservable reach set by 𝒓𝒕 = 𝒓𝒕 ∪ 𝒚, ℓ 𝒚 ∈ 𝜺 𝒚𝟏, 𝒗 , 𝒗 ∈ 𝜯𝒗𝒑

∗ , ℓ = 𝜶

𝑶 , 𝒗 where ∇ is a labeling function. iii. Next, 𝒓𝒕 is recursively extended to all of the plant’s reachable states. iv. The diagnoser’s initial state 𝒓𝟏 ⊆ 𝒓𝒕 can be obtained as 𝒓𝟏 = 𝒚, ℓ 𝒚, ℓ ∈ 𝒓𝒕, 𝒚 ∈ 𝑽𝑺(𝒀𝒕) . Step 2: Constructing remaining diagnoser states 𝒓 ∈ 𝑹𝒆 Starting with 𝒓𝟏, Step 2 constructs the remaining accessible states of the diagnoser 𝒓 ∈ 𝑹𝒆 by 𝜺𝒆 𝒓, 𝒇 =

𝒚,ℓ ∈𝒓 𝒖∈𝑽𝑭 𝒇,𝒚

{ 𝜺 𝒚, 𝒇 , 𝜶 ℓ, 𝒖 } Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 24

• A. White, A. Karimoddini, “Semi-Asynchronous Fault Diagnosis of Discrete Event Systems,” The 2016 IEEE International Conference on Systems, Man, and Cybernetics.

Algorithmic Construction: Example

Diagnoser?

Starting with the Original System, we will now construct its Semi-Asynchronous Diagnoser

Algorithm

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 25 𝒀𝒕 = 𝟑, 𝟘 𝜯𝒑 = 𝜷, 𝜸, 𝜺 𝜯𝒗 = {𝒈}

Original System

Step 1: Constructing 𝑟0

I. 𝒓𝒕 = 𝒚𝟏, 𝑶 . II. 𝒓𝒕 = 𝒓𝒕 ∪ 𝒚, ℓ 𝒚 ∈ 𝜺 𝒚𝟏, 𝒗 , 𝒗 ∈ 𝜯𝒗𝒑

∗ , ℓ = 𝜶

𝑶 , 𝒗

• III. 𝒓𝒕 is recursively extended to all of the plant’s reachable states

IV. 𝒓𝟏 = 𝒚, ℓ 𝒚, ℓ ∈ 𝒓𝒕, 𝒚 ∈ 𝑽𝑺(𝒀𝒕) . I. 𝒓𝒕 = 𝟐, 𝑶 II. 𝒓𝒕 = { 𝟐, 𝑶 , (𝟔, 𝑮)}

• III. 𝒓𝒕 = { 𝟐, 𝑶 , 𝟔, 𝑮 , 𝟑, 𝑶 , 𝟒, 𝑶 , 𝟓, 𝑮 , 𝟔, 𝑮 , 𝟕, 𝑮 ,

𝟖, 𝑶 , 𝟗, 𝑮 , 𝟘, 𝑮 , (𝟐𝟏, 𝑮)} IV. 𝒓𝟏 = { 𝟑, 𝑶 , (𝟘, 𝑮)}

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 26 𝒀𝒕 = 𝟑, 𝟘 𝜯𝒑 = 𝜷, 𝜸, 𝜺 𝜯𝒗 = {𝒈}

Original System

Step 2: Constructing Diagnoser States 𝑟 ∈ 𝑅𝑒

Starting with 𝒓𝟏, construct the remaining accessible states of the diagnoser 𝒓 ∈ 𝑹𝒆 by

𝜺𝒆 𝒓, 𝒇 =

𝒚,ℓ ∈𝒓 𝒖∈𝑽𝑭 𝒇,𝒚

{ 𝜺 𝒚, 𝒇 , 𝜶 ℓ, 𝒖 } 𝒀𝒕 = 𝟑, 𝟘 𝜯𝒑 = 𝜷, 𝜸, 𝜺 𝜯𝒗 = {𝒈}

Original System

𝒓𝟏 = 𝟑, 𝑶 , (𝟘, 𝑮) 𝒓𝟐 = { 𝟒, 𝑶 , 𝟐𝟏, 𝑮 , 𝟖, 𝑶 , (𝟕, 𝑮)} 𝒓𝟑 = { 𝟐, 𝑶 , 𝟗, 𝑮 , 𝟔, 𝑮 , } 𝒓𝟒 = {(𝟓, 𝑮)}

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 27

Algorithmically Constructed Diagnoser 𝐻𝑒

𝐇𝐞 = 𝐑𝐞, 𝚻𝐞, 𝛆𝐞, 𝐫𝟏 a four-tuple representation of diagnoser generated by the algorithm

Algorithm

Diagnoser

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 28 𝒀𝒕 = 𝟑, 𝟘 𝜯𝒑 = 𝜷, 𝜸, 𝜺 𝜯𝒗 = {𝒈}

Original System

Simulation

𝛽 𝛽 𝛽 𝜀 𝜀 𝑔 𝑔 𝑔 𝜀 𝛾 𝛽 𝛾 𝛽

𝑌𝑡 = 2,9 Σ𝑝 = 𝛽, 𝛾, 𝜀 Σ𝑣 = {𝑔}

q0 = { 2, N , (9, F)} q1 = { 3, N , 10, F , 7, N , (6, F)} q2 = { 1, N , 8, F , (5, F)} q3 = {(4, F)}

𝛾 𝜀 𝛽 𝛽 𝛽 The original system is in State 2, however, only having access to 𝒀𝒕, the diagnoser initially estimates the system to be in States 2 or 9

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 29

Simulation

𝛽 𝛽 𝛽 𝜀 𝜀 𝑔 𝑔 𝑔 𝜀 𝛾 𝛽 𝛾 𝛽

𝑌𝑡 = 2,9 Σ𝑝 = 𝛽, 𝛾, 𝜀 Σ𝑣 = {𝑔}

q0 = { 2, N , (9, F)} q1 = { 3, N , 10, F , 7, N , (6, F)} q2 = { 1, N , 8, F , (5, F)} q3 = {(4, F)}

𝛾 𝜀 𝛽 𝛽 𝛽

System Trace Observed System Trace Natural Projection 𝑄: Σ∗ → Σ𝑝

∗

2→

𝛾 3

𝑟0 →

𝛾 𝑟1

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 30

Simulation

𝛽 𝛽 𝛽 𝜀 𝜀 𝑔 𝑔 𝑔 𝜀 𝛾 𝛽 𝛾 𝛽

𝑌𝑡 = 2,9 Σ𝑝 = 𝛽, 𝛾, 𝜀 Σ𝑣 = {𝑔}

q0 = { 2, N , (9, F)} q1 = { 3, N , 10, F , 7, N , (6, F)} q2 = { 1, N , 8, F , (5, F)} q3 = {(4, F)}

𝛾 𝜀 𝛽 𝛽 𝛽

System Trace Observed System Trace

2→

𝛾 3→ 𝑔 6

𝑟0 →

𝛾 𝑟1

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 31 Natural Projection 𝑄: Σ∗ → Σ𝑝

∗

Simulation

𝛽 𝛽 𝛽 𝜀 𝜀 𝑔 𝑔 𝑔 𝜀 𝛾 𝛽 𝛾 𝛽

𝑌𝑡 = 2,9 Σ𝑝 = 𝛽, 𝛾, 𝜀 Σ𝑣 = {𝑔}

q0 = { 2, N , (9, F)} q1 = { 3, N , 10, F , 7, N , (6, F)} q2 = { 1, N , 8, F , (5, F)} q3 = {(4, F)}

𝛾 𝜀 𝛽 𝛽 𝛽

System Trace Observed System Trace

2→

𝛾 3→ 𝑔 6→ 𝜀 8

𝑟0 →

𝛾 𝑟1→ 𝜀 𝑟2

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 32 Natural Projection 𝑄: Σ∗ → Σ𝑝

∗

Simulation

𝛽 𝛽 𝛽 𝜀 𝜀 𝑔 𝑔 𝑔 𝜀 𝛾 𝛽 𝛾 𝛽

𝑌𝑡 = 2,9 Σ𝑝 = 𝛽, 𝛾, 𝜀 Σ𝑣 = {𝑔}

q0 = { 2, N , (9, F)} q1 = { 3, N , 10, F , 7, N , (6, F)} q2 = { 1, N , 8, F , (5, F)} q3 = {(4, F)}

𝛾 𝜀 𝛽 𝛽 𝛽

System Trace Observed System Trace

2→

𝛾 3→ 𝑔 6→ 𝜀 8→ 𝛽 9

𝑟0 →

𝛾 𝑟1→ 𝜀 𝑟2→ 𝛽 𝑟0

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 33 Natural Projection 𝑄: Σ∗ → Σ𝑝

∗

Simulation

𝛽 𝛽 𝛽 𝜀 𝜀 𝑔 𝑔 𝑔 𝜀 𝛾 𝛽 𝛾 𝛽

𝑌𝑡 = 2,9 Σ𝑝 = 𝛽, 𝛾, 𝜀 Σ𝑣 = {𝑔}

q0 = { 2, N , (9, F)} q1 = { 3, N , 10, F , 7, N , (6, F)} q2 = { 1, N , 8, F , (5, F)} q3 = {(4, F)}

𝛾 𝜀 𝛽 𝛽 𝛽

System Trace Observed System Trace

2→

𝛾 3→ 𝑔 6→ 𝜀 8→ 𝛽 9→ 𝛽 4

𝑟0 →

𝛾 𝑟1→ 𝜀 𝑟2→ 𝛽 𝑟0→ 𝛽 𝑟3

Diagnoser reaches and remains at 𝑟3, an Fi − certain state

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 34 Natural Projection 𝑄: Σ∗ → Σ𝑝

∗

Are All System Faults Diagnosable?

The proposed algorithm will create a Semi-Asynchronous Diagnoser 𝐻𝑒 for any provided system 𝐻 Will the created Semi-Asynchronous Diagnoser 𝐻𝑒 diagnose all modelled system fault occurrences

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 35

Fault Occurrence Does Not Lead To 𝐺

𝑗 −Certain State

System Trace Observed System Trace Natural Projection 𝑄: Σ∗ → Σ𝑝

∗

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 36

𝐺

𝑗 −Semi-Asynchronous Diagnosability

Definition: The plant 𝐇 with the live language 𝓜𝐇, is said to be 𝑮𝒋-semi-asynchronously diagnosable with respect to the failure type 𝚻𝒈𝐣, the natural projection 𝑸, and the initial set

• f estimated system states 𝐘𝐓, if and only if

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 37

CASE 1 CASE 2

Example: Semi-Asynchronous Diagnosability

𝒈𝟐 𝜷 𝒈𝟑 𝜸 𝜷 𝜸 𝜺 𝜸 𝜷 𝜺 𝜷 𝜷 𝜺 𝜸 𝜺 𝜷 𝜸 𝜷 𝜷 𝜺 𝜸 𝒈𝟐 𝒈𝟑 𝜷 𝜸 𝜺 𝜷

𝑡1 = 𝛽, 𝑡2 = 𝛽𝛾𝑔

2𝛽𝜀

𝑡1. 𝑡2 = 𝛽𝛽𝛾𝑔

2𝛽𝜀 ∈ ℒ𝐻

𝑡3 ∈ ℒ𝐻/𝑡 1 .𝑡2 = 𝛾𝛽𝜀 ∗

Upon reaching state 𝒚 = 𝟑, the system successively follows the sequence

*: arbitrarily repeated sequence

𝑣 = 𝛽, 𝑤 = 𝛽𝛾 𝛽𝜀𝛾 ∗ 𝑣. 𝑤 ∈ ℒ𝐻 𝑣 ∈ 𝑄𝑠𝑓 𝑌𝑡 , 𝑤 ∈ 𝑄𝑓𝑦𝑢 ℒ𝐻

−1

𝑄 𝑡2. 𝑡3  𝑔

2 ∉ 𝑣𝑤 ⇒ 𝐺 𝑗-Semi-Asynchronous

Diagnosability is VIOLATED for CASE 1  ⇒ 𝐻2 is not 𝐺

2-Semi-Asynchronously

Diagnosable with respect to 𝑌𝑡

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 38

Conclusion

 In this presentation, we have introduced a novel diagnoser that may be

activated asynchronously while the system under diagnosis is in operation.

 Provided an algorithm for constructing the diagnoser  Introduced the concept of Semi-Asynchronous Diagnosability, in a formal

definition

 Provided the conditions for Semi-Asynchronous Diagnosability with respect to

diagnoser activation

 Relative to a set of estimated system state locations 𝒀𝒕; the constructed

diagnoser is capable of diagnosing fault occurrences in the system without access to information on system operation prior to the system reaching 𝒀𝒕

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 39

Future Work

 Develop systematic methods to verify the semi-asynchronous diagnosability

• f a given system

 Extension to distributed and/or decentralized architectures for scalability

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 40

Acknowledgements

 My teammates at ACCESS Laboratory and

TECHLAV Center

 Financial support from Title III  Support from Air Force Research Laboratory and

Office of the Secretary of Defense under agreement number FA8750-15-2-0116

 Support from US ARMY Research Office under

agreement number W911NF-16-1-0489

http://accesslab.net/ http://techlav.ncat.edu/ Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 41

Questions

Semi-asynchronous Fault Diagnosis of Discrete Event Systems Alejandro White 42