On-time diagnosis of discrete event systems Aditya Mahajan and Demosthenis Teneketzis Dept. of EECS, University of Michigan, Ann Arbor, MI. USA. WODES 2008, May 30, 2008.
Fault Diagnosis in DES 1. Asymptotic (accuracy is critical; delay is important but not critical) 2. On-time (delay is critical; accuracy is important but not critical) Most of the literature on diagnosis of DES has concentrated on asymptotic fault diagnosis. Contribution of this paper • Formulate on-time fault diagnosis as a minimax optimization problem. • Use decision theory to provide a solution methodology.
Modelling questions • What do we mean by “time”? • What should the diagnoser/monitor do? • How do we model performance? When it is time to take a decision but the monitor is not sure that a fault has occurred, it will make mistakes.
Preliminaries Language, Monitor, and Costs
Language • Language L is prefix-closed, finite, and bounded L = L T ∪ L NT • Terminal Strings: L T := { s ∈ L : L \ s � = ∅ } • Non-terminal Strings: L NT := L \ L T . • Event Set Σ = Σ o ∪ Σ uo = ⇒ natural projections. • • Observable events: Σ o Unobservable events: Σ uo . • Fault event f ∈ Σ uo .
Monitor • Observes P ( L ) • Upon observing an event, the monitor can: ◦ raise an alarm , = ⇒ the system is shut down immediately. ◦ ⇒ the system continues to operate. do nothing , = • Monitoring policy g : P ( L ) → { 0, 1 } • Monitored sub-language L | g Sub-language where the system can stop ⇒ system stops in L S NT ∪ L S • Monitor raises an alarm = T L S L S NT = { s · σ ∈ L NT : σ ∈ Σ o } , T = { s · σ ∈ L T : σ ∈ Σ o } • Monitor does not raise an alarm = ⇒ system stops in L T System can stop in L S = L S For any g , ( L | g ) T ⊆ L S • NT ∪ L T •
Example e a e a a f a f b a b a a d d d d b a b a b a d d d d d b a b a b a e e e e d b a b a b a d d d d d b a b a b a Language L P ( L ) L | g for g ( add ) = 1
Quantifying timeliness • After a fault has occurred, each event incurs a cost c . • ⇒ false alarm penalty of H NT . System is stopped in a non-faulty state = • ⇒ System executes a terminal trace in a faulty state = additional terminal penalty of H T . Cost of stopping • For s ∈ L , let ◦ τ ( s ) be the first stage when a fault occurs in s . ◦ n be the “length” of s � ( n − τ ( s )) c, if s contains a fault, for s ∈ L S • C ( s ) = NT , H NT , otherwise; � ( n − τ ( s )) c + H T , if s contains a fault, • for s ∈ L T , C ( s ) = otherwise. 0,
Problem Formulation
The on-time diagnosis problem • Given ◦ Prefix-closed, finite, and bounded language L , ◦ Observable events Σ o , unobservable events Σ uo , and fault event f ◦ Cost c , fault alarm penalty H NT , and a terminal penalty H T . • Define ◦ G family of functions from P ( L ) to { 0, 1 } ◦ Performance of a monitoring policy g ∈ G J ( g ) := max C ( s ) . s ∈ ( L | g ) T • Choose A monitoring rule g ∗ ∈ G to minimize J ( g ) ◦ J ∗ = J ( g ∗ ) = min max C ( s ) g ∈ G s ∈ ( L | g ) T
Centralized minimax optimization problem Can be solved by dynamic programming
Some Notation Q ( t ) := { s · σ ∈ P − 1 ( t ) : σ ∈ Σ o } • Q T ( t ) := P − 1 ( t ) ∩ L T • Optimal monitoring rule • For t ∈ ( P ( L )) T � � V ( t ) = min s ∈ Q ( t ) C ( s ) max s ∈ Q T ( t ) C ( s ) max , minimum worst case cost worst case cost worst case of stopping of continuing cost to go at t • For t ∈ ( P ( L )) NT , let O C ( t ) := { e ∈ Σ : t · e ∈ P ( L ) } , and � � � � e ∈ O C ( t ) V ( t · e ) V ( t ) = min s ∈ Q ( t ) C ( s ) max , max s ∈ Q T ( t ) C ( s ) , max max minimum worst case cost worst case cost of continuing worst case of stopping cost to go at t
Example e a ǫ a f a ea ( H NT ) , b a a ( H NT ) d d b a d ead ( H NT ) , b a eab ( 0 ) afa ( c + H T ) d d afd ( c ) b a d b a e e eadd ( H NT ) , eadb ( 0 ) afda ( 2c + H T ) afdd ( 2c ) b a d b a d d eaddb ( 0 ) , eadded ( H NT ) , afdda ( 3c + H T ) , eaddeb ( 0 ) afdded ( 4c ) afddea ( 4c + H T ) b a b a eaddedb ( 0 ) afddeda ( 5c + H T ) Language L Optimal monitor for H T = c , H NT = 3c
Relaxing some modelling assumptions • Live languages Should be possible. Working on the details. • Generalized costs Use a trace dependent cost in the paper • Generalized projections Use prefix-preserving projections in the paper Summary • Formulate and solve on-time fault diagnosis problem. • Penalize false alarm and (trace dependent) amount of delay in fault detection. • Equivalent to a minimax optimization problem.
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