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Semi-asynchronous Fault Diagnosis of Discrete Event Systems ALEJANDRO WHITE DR. ALI KARIMODDINI OCTOBER 2017 NC A&T State University # Alejandro White Semi-asynchronous Fault Diagnosis of Discrete Event Systems http://techlav.ncat.edu/ http://www.ncat.edu/ http://accesslab.net/

Table Of Contents  Motivation  Background  Problem Statement  System Under Diagnosis  Diagnoser  Semi-Asynchronous Diagnosability  Simulated Example  Conclusion 2 Alejandro White Semi-asynchronous Fault Diagnosis of Discrete Event Systems

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

Increase of roles and presence in daily societal activities leads to an increase in liability Crash Report Perrone Robotics driverless car crashes after being hacked during testing. Alejandro White Semi-asynchronous Fault Diagnosis of Discrete Event Systems 4

Failure to properly diagnose faults, leads to incorrect recovery actions American Airlines Flight 191 (1979) • Left Engine separated from wing • Pilot only 15s to react • Subsequent analysis shows consequence of faults avoidable Alejandro White Semi-asynchronous Fault Diagnosis of Discrete Event Systems 5

General Objectives & Impacts of Fault Diagnosis Impacts Objectives Upon a fault occurrence, a system will To develop systematic techniques for automatic autonomously become aware of the fault’s diagnosis of faults in the system to timely diagnose occurrence, and initiate a systematic procedure that (detect, identify and locate) occurred system faults. locates, identifies, and accommodates the fault in order to ensure proper utilization of the system’s remaining resources, allowing for a resilient post fault system operation that is both safe and stable. Alejandro White Semi-asynchronous Fault Diagnosis of Discrete Event Systems 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 operation; in turn highlighting faulty behaviors of the system. Alejandro White Semi-asynchronous Fault Diagnosis of Discrete Event Systems 7

Preliminaries and Background

Automaton  Definition: a non-deterministic finite-state Discrete-Event System (DES) can be represented by a four-tuple: 𝐇 = (𝐘, 𝚻, 𝛆, 𝒚 𝟏 ) 𝛽 𝛾  State space ( 𝒀 ): a discrete set of system states  Event set ( 𝜯 = 𝜯 𝒑 ⋃𝜯 𝒗 ): notable occurrences of asynchronous discrete 𝑣 𝛽 changes in a system  Observable events ( 𝜯 𝒑 ): events observed by a sensor (e.g., flowing of  𝒀 = {𝟐, 𝟑, 𝟒} water)  𝚻 = {𝐯, 𝜷, 𝜸}  Unobservable events ( 𝜯 𝒗 ): events that are unable to be detected by  𝚻 𝒑 = {𝜷, 𝜸} sensors; possibly due to sensor absence/damage (e.g., failure event)  𝚻 𝒗 = {𝒗}  𝜺 𝟐, 𝒗 = 𝟑 , 𝜺 𝟑, 𝜷 = {𝟑, 𝟒} ,  State-transition relation ( 𝜺: 𝒀 × 𝜯 → 𝟑 𝒀 ): a partial relation that determines all 𝜺 𝟒, 𝜸 = 𝟒 feasible system state transitions caused by system events ( 𝟑 𝒀 is the set of all  𝒚 𝟏 = {𝟐} possible combinations of states)  Initial state ( 𝒚 𝟏 ): indicated by an input arrow connected to a single state Alejandro White Semi-asynchronous Fault Diagnosis of Discrete Event Systems 9

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 𝛾  Trace (string): a sequence of one or more events, allowable by the system’s behavior 𝑣 𝛽  e.g., 𝒕 = 𝒇 𝟐 𝒇 𝟑 … 𝒇 𝒐 where 𝒇 𝒋 ∈ 𝜯  ℒ 𝑯 𝒚 𝟏  Language ( 𝓜 𝑯 (𝒚 𝟏 ) ): the set of all system traces = {𝜻, 𝒗, 𝒗𝜷 ∗ , 𝒗𝜷 ∗ 𝜷, 𝒗𝜷 ∗ 𝜸 ∗ , 𝒗𝜷 ∗ 𝜸𝜸 ∗ , … } = {𝜻, 𝒗𝜷 ∗ 𝜸 ∗ } which originate at the system’s initial state 𝒚 𝟏  𝓜 𝑯 (𝒚 𝟏 ) = {𝒕 ∈ 𝚻 ∗ |𝜺 𝒚 𝟏 , 𝒕 𝐣𝐭 𝐞𝐟𝐠𝐣𝐨𝐟𝐞} *: arbitrarily repeated string ( 𝚻 ∗ is the Kleene Closure of 𝚻 ) Alejandro White Semi-asynchronous Fault Diagnosis of Discrete Event Systems 10

Natural Projection  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. 𝛽 𝛾 𝑸: 𝚻 ∗ → 𝚻 𝒑 ∗ 𝑸 𝜻 = 𝜻 𝑣 𝛽 𝑸 𝒇 = 𝒇 if 𝒇 ∈ 𝚻 𝟏 𝑸 𝒇 = 𝜻 if 𝒇 ∉ 𝚻 𝒑  ℒ 𝑯 𝒚 𝟏 = {𝒗, 𝒗𝜷 ∗ 𝜸 ∗ } for 𝒕 ∈ 𝚻 ∗ and 𝒇 ∈ 𝚻 𝑸(𝒕𝒇) = 𝑸 𝒕 𝑸 𝒇  𝐐(ℒ 𝑯 𝒚 𝟏 ) = {𝜻, 𝜷 ∗ 𝜸 ∗ } Extension of the natural projection to the languages: −1  𝑄 𝜷 = {𝒗𝜷} ℒ 𝑯 (𝒚 𝟏 ) 𝑄 ℒ 𝐇 (𝐲 𝟏 ) = {𝑄 𝑡 ∣ 𝑡 ∈ ℒ 𝐇 (𝐲 𝟏 )} Inverse of Natural Projection −1 𝑄 𝑥 = {𝑡 ∈ ℒ 𝐡 (𝐲 𝟏 ) ∣ 𝑄 𝑡 = 𝑥} ℒ 𝑯 (𝒚 𝟏 ) Alejandro White Semi-asynchronous Fault Diagnosis of Discrete Event Systems 11

System String Sets  Here we present pre-defined sets of system strings ≔ {𝒘 ∈ 𝚻 ∗ ∣ ∃𝒗 ∈ ℒ 𝑯 𝒚 𝟏 : 𝒗𝒘 ∈ ℒ 𝑯 (𝒚 𝟏 )}  Extension Closure: 𝐟𝐲𝐮 ℒ 𝑯 𝒚 𝟏  𝑸𝒔𝒇 𝒀 𝒕 : 𝑸𝒔𝒇 𝒀 𝒕 = {𝒕 ∈ ℒ 𝑯 𝒚 𝟏 ∣ 𝜺 𝒚 𝟏 , 𝒕 ∈ 𝒀 𝒕 } , the set of strings leading to 𝒀 𝒕 generated from 𝒚 𝟏  𝑸𝒑𝒕𝒖 𝒀 𝒕 : the set of strings that can be generated from any 𝒚 ∈ 𝒀 𝒕 ∗ , 𝜺 𝒚, 𝒗 = 𝒛} , the set of all system states (with the inclusion of x itself)  Unobservable Reach: 𝑽𝑺 𝒕 = {𝒛 ∈ 𝒀 ∣ ∃𝒗 ∈ 𝚻 𝐯 reachable from state 𝒚 via strings solely consisting of unobservable events ∗ 𝒃𝒐𝒆 𝒕. 𝒖 ∈ ℒ 𝑯 (𝒚)} , specifies the set of all unobservable extensions of 𝒕  Unobservable Extension: 𝑽𝑭 𝒕, 𝒚 = {𝒕. 𝒖 ∣ 𝒖 ∈ 𝚻 𝒗 concatenated with the string 𝒕 , and generated from the state x Alejandro White Semi-asynchronous Fault Diagnosis of Discrete Event Systems 12

Diagnoser Functionality 𝓜 𝑯 (𝒚 𝟏 ) 𝑸(𝓜 𝑯 (𝒚 𝟏 )) Fault Detection Natural Plant Fault Isolation Projection Diagnoser 𝑯 = (𝒀, 𝚻, 𝜺, 𝒚 𝟏 ) Fault 𝑸: 𝚻 ∗ → 𝚻 𝒑 ∗ 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 occurrences.  Upon observance of the original system’s behavior, the diagnoser updates its estimation of the original system’s state and condition. Alejandro White Semi-asynchronous Fault Diagnosis of Discrete Event Systems 13

DES Fault Diagnosis (State-Of-The-Art) STRATEGY Event-based: Sampath et al.1995 TOOLS STRUCTURE State-based: Lin 1994, Zad et al. 2003 Petri Nets: Prock 1991, Basile Centralized: Sampath et al. et al. 2008 1995, Zad et al. 2003 Process Algebra: Console et al. Decentralized: Wang et al. 2002 2007, Lafortune et al. 2005 Automata Theory: Sampath Distributed: Fabre 2002, 1995, Wang et al., 2007 Pencolé 2005 Fault Diagnosis Alejandro White Semi-asynchronous Fault Diagnosis of Discrete Event Systems 14

Bridging Gaps In DES Diagnosis Methods In many cases, In all of existing methods, the it is not possible, or it is diagnoser has to be time-consuming and costly, simultaneously initialized to restart the plant to be with the system under synchronized with the diagnosis. diagnoser. In all of existing methods, the In many practical situations, diagnoser should synchronously only after a fault occurs, the execute the events in parallel with diagnosis tool can be brought in system under diagnosis, to keep and connected to the faulty the past history of exhibited plant to diagnose the occurred normal and faulty behaviors. fault. Alejandro White Semi-asynchronous Fault Diagnosis of Discrete Event Systems 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). Alejandro White Semi-asynchronous Fault Diagnosis of Discrete Event Systems 16