Diagnosis with Fault Modes

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Diagnosis with Fault Modes

• Philippe Dague and Yuhong Yan
• NRC-IIT
• Philippe.dague_at_lipn.univ-paris13.fr
• Yuhong.yan_at_nrc.gc.ca

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Diagnosis Using fault modes

• MBD approach provides a framework for diagnosing
  (detecting and locating faults) a device from its
  correct behavioural model only.
• Big advantage w.r.t. techniques based on a priori
  knowledge of all failure modes.
• Nevertheless, knowledge of likely faulty modes
  increases discrimination capacity and may allow
  fault identification.
• Idea extending the consistency-based diagnostic
  framework by using fault modes, but without
  requiring their exhaustivity.
• Sherlock (de Kleer Williams) and GDE (Struss
  Dressler) in 1989.

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Behavioural modes

• In addition to good behaviour G(c), we consider
  known faulty behaviours Fi(c) and unknown mode
  U(c), all distinct.
• U(c) is used to model the non exhaustivity of the
  Fis, keeping logical soundness.
• Example of an inverter
• Inverter(C) ? AB(C) ? G(C)
• Inverter(C) ? AB(C) ? S0(C) ? S1(C) ? U(C)
• Inverter(C) ? G(C) ? S0(C)
• Inverter(C) ? S1(C) ? U(C)
• G(C) ? (In(C)0 ? Out(C)1) ? (In(C)1 ?
  Out(C)0)
• S0(C) ? Out(C)0
• S1(C) ? Out(C)1

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Definition of Diagnosis and Conflict

• Diagnosis is an assignment of behavioural mode to
  each component, consistent with system
  description and observations
• SD?OBS?mi(c)c?COMPONENTS ? ?
• Diagnoses can still be computed from minimal
  conflicts
• A conflict is a set of component behavioural
  modes, inconsistent with system description and
  observations
• SD?OBS?mi(ck) ?
• Minimal conflicts minimal for set inclusion
• Can be computed as nogoods by an ATMS
  (assumptions components behavioural modes)

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Computation of Diagnoses

• An assignment ? of behavioural mode to each
  component is a diagnosis iff it does not contain
  any minimal conflict Ci.
• That is, complement of ? in ?modes(c) c ?
  COMPONENTS is a hitting set of the collection of
  minimal conflicts Ci.
• Example the 3-inverter

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The 3-inverter
A
B
C
X
Y
O
I
Space of Diagnoses U 4364
ATMS assumptions 34 12 G(A),S0(A),S1(A),
U(A), G(B), S0(B),
Comparison n components, diagnoses space U with
only good modes 2n with k faulty modes, 1 good
mode, 1 unknown mode (k2)n
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The 3-inverter ATMS label computation (1)
Observation I0
I0,
X1,G(A),S1(A)
X0,S0(A)
Y0,G(A), G(B),S1(A),G(B),S0(B)
Y1,S0(A),G(B),S1(B)
O1,G(A), G(B), G(C), S1(A),G(B), G(C),
S0(B), G(C), S1(C)
O0,S0(A),G(B),G(C),S1(B),G(C),S0(C)
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The 3-inverter ATMS label computation (2)
Observation I0, O0
I0,
X1,G(A),S1(A)
X0,S0(A), G(C),G(B)
Y0,G(A), G(B),S1(A),G(B),S0(B)
Y1,S0(A),G(B),S1(B), G(C)
O1,G(A), G(B), G(C), S1(A),G(B), G(C),
S0(B), G(C), S1(C)
O0,S0(A),G(B),G(C),S1(B),G(C),S0(C)
O0,
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The 3-inverter 4 minimal conflicts

• 4 minimal conflicts Ci G(A), G(B),
  G(C),S1(A),G(B), G(C), S0(B), G(C), S1(C)
• Diagnoses assign modes to the components,
  ?mi(A), mj(B), mk(C)
• Diagnoses do not contain any of the 4 minimal
  conflicts
• ?Ci, (Ci ? ?) ? Ci ? (U\?)??
• ? (U\?) hits any Ci
• ? (U\?) is hitting set of conflicts

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The 3-inverterhitting sets

• The hitting sets of 4 minimal conflicts Ci
  G(A), G(B), G(C),S1(A),G(B), G(C), S0(B),
  G(C), S1(C) are
• G(C),S1(C)
• G(B),S0(B),S1(C)
• G(A),S1(A),S0(B),S1(C)

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The 3-inverter diagnoses
From hitting set G(C),S1(C) mi(A),mj(B),S0(C)
or mi(A),mj(B),U(C)
From hitting set G(B),S0(B),S1(C) mi(A),S1(B),
G(C) or mi(A),U(B),G(C)
From hitting set G(A),S1(A),S0(B),S1(C) S0(A),
G(B),G(C) or U(A),G(B),G(C)
Total 42 diagnoses out of 64 in diagnoses space
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The 3-inverter compare with using only good mode
G() ? AB() U() ? AB()
For the diagnoses mi(A),mi(B),S0(C) ?
X mi(A),mi(B),U(C) ? G(A),G(B),U(C) ?
AB(A),AB(B),AB(C) ? C
G(A),U(B),U(C) ? AB(A),AB(B),AB(C) ? B,C
U(A),G(B),U(C) ? AB(A),AB(B),AB(C) ?
A,C
U(A),U(B),U(C) ? AB(A),AB(B),AB(C) ? A,B,C
mi(A),S1(B),G(C) ? X mi(A),U(B),G(C) ?
G(A),U(B),G(C) ? AB(A),AB(B),AB(C) ? B
U(A),U(B),G(C) ?
AB(A),AB(B),AB(C) ? A,B
S0(A),G(B),G(C) ? X U(A),G(B),G(C) ?
AB(A),AB(B),AB(C) ? A
minimal diagnoses are underlined
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The Probability of Diagnoses

• Diagnoses space is large even for a small problem
  when the faulty modes are considered
• The complete diagnoses are able to get but not
  necessary for practical work
• Only the most probable diagnoses are needed to be
  found the leading diagnoses (a subset of all the
  diagnoses)

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The criteria of the leading diagnoses

• All leading diagnoses have higher probability
  than all non-leading diagnoses
• Select no more than k1(5) leading diagnoses
• Probability gt Max(pi)/k2
• ?(pi)gtk3, k30.75, stop to select pi when the sum
  is greater than k3

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Focus in ATMS

• Focuses are the leading diagnoses
• Only consider the environments ? one of the
  focuses
• Best first search to get the focus select the
  most probable diagnoses

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Probabilities of Diagnoses

• Given prior probability p(mi(ck))
• The prior probability of a diagnosis ?i is
• p(?i) ?mj??i p(mi(ck))
• Update the posterior probability after a new
  measurement
• p(?ixivik) p(xivik?i)p(?i)/p(xivik)
• p(xivik?i) 0 if ?i ?Rik
• 1 if ?i ?Sik
• 1/m if ?i ?Ui
• p(xivik) p(Sik) p(Ui)/m

Rik Sik? Ui Sikxivik Ui xi? Others
x?vik
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The cost of measurements

• Shannon entropy
• H - ?pilogpi
• The expected entropy He(xi) after measure
  quantity xi is
• He(xi) - ?p(xivik)H(xivik)
• H(xivik)
• H(xivik) - ?pllogpl
• He(xi) H?p(xivik)logp(xivik)p(Ui)logm
• (xi) ?p(xivik)logp(xivik)p(Ui)logm1

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Simplified Idea

• Assume components fail independently with equal
  very small probability ?
• p(?i) ??i(1- ?)n-?i? ??i
• ?i the number of components in ?i
• After multiple probe E,
• p(?iE) ?q /(Nmfl)
• flnumber of times ?i failed to predict a
  measurement outcome in the sequence diagnosis
• qsize of ?i N normalization
• If ? ltlt1/mfl, we can keep only minimal
  cardinality diagnoses and N ?q ?(1/mfl)
• p(?iE) 1/mfl ?(1/mfl)
• It is independent from ?

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Cost of Measurements

• (xi) is to minimize ?p(Sik)p(Ui)/mlogp(Sik)
  p(Ui)/mp(Ui)logm
• (??p(xivik)logp(xivik)p(Ui)logm1)
• This function is independent of ? and depends
  only of fl
• Special case diagnoses (of minimal cardinality)
  always predict outcomes gt p(Ui)0 and fl 0 gt
• P(?iE) 1/minimal cardinality diagnoses
  1/N
• Minimize cost
• ?p(Sik)logp(Sik) ?(Cik/N)log(Cik/N)
• ?minimize ?(Cik)log(Cik)
• Where Cik number of diagnoses (in N)
  predicts that xivik

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Example ploybox

• 2 single fault diagnoses M1 and A1
• M1?SD?OBSX4,Y6,Z6
• A1?SD?OBSX6,Y6,Z6
• (X) 1ln11ln10, the best!
• (Y) 2ln2 1.4
• (Z) 2ln2 1.4