Fundamentals of ModelBased Diagnosis

1
Fundamentals of Model-Based Diagnosis

• Johan de Kleer
• James Kurien

Safe Process 6/11/2003
2
Basic Underlying Assumptions

• Physical system
• comprised of components
• desired function
• design achieves function
• system is correct instance of design
• All malfunctions caused by faulty component(s)
• Behavioral information
• only indirect evidence

3
Model-Based DX Tasks

• Detect faulty system behavior
• Identify the faulty system components
• Identify additional system evidence
• more observations (troubleshooting)
• change inputs
• Repair the system
• Reconfigure the system
• FMEA construction
• Repair manual production
• Write embedded controller software

4
Observations
Replace R6
Diagnoses Repairs Actions
Design structural description
Diagnostic Reasoner
Domain Knowledge Component Models
5
Rule-based DX

• Predefined set of possible faults
• Predefined set of possible symptoms
• Predefined relations among them
• Does not generalize
• Is not robust

If gram stain of the organism is gram negative,
and the morphology of the organism is rod, and
the aerobicity of the organism is anaerobic, then
there is evidence (.7) that the identify of the
organism is Bacteriodes Mycin.
6
Achievements of Model-Based DX
7
Speed up due to new algorithms
Slowdowns due to additional inferences
Performance
SingleFaults Multiple Faults Fault Modes
Probing Dynamics Control Hybrid Systems
Coverage
8
A Simple Expository Example
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A Simple Expository Example
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A Simple Expository Example
X
6
F
M1
A1
Y
M2
G
A2
Z
M3
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A Simple Expository Example
X
6
F
M1
A1
6
Y
M2
G
A2
Z
M3
12
A Simple Expository Example
X
6
F
F
M1
A1
12
6
Y
M2
G
A2
Z
M3
13
A Simple Expository Example
X
6
F
F
M1
A1
12
6
Y
M2
G
A2
Z
M3
6
14
A Simple Expository Example
X
6
F
F
M1
A1
12
6
Y
M2
G
A2
12
Z
M3
6
15
A Simple Expository Example
X
6
10
F
F
M1
A1
12
6
Y
M2
12
G
A2
12
Z
M3
6
16
Observations
Design structural description
Diagnostic Reasoner
Diagnoses
A
Compositional NFIS
Domain Knowledge Component Models
17
Constraint Suspension
X
6
10
F
F
M1
12
6
Y
M2
12
G
A2
12
Z
M3
6
18
Constraint Suspension
X
6
10
F
F
M1
A1
6
Y
M2
12
G
A2
12
Z
M3
6
Remove constraints for A1 eliminates the
discrepency, so A1 is a diagnosis.
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Constraint Suspension
X
6
10
F
F
M1
A1
6
Y
M2
12
G
A2
12
Z
M3
6
Remove constraints for A1 eliminates the
discrepency, so A1 is a diagnosis.
20
Constraint Suspension
X
6
10
F
F
M1
A1
12
6
Y
M2
12
G
A2
12
Z
M3
6
Removing constraint for M2 does not eliminate
the discrepency, so M2 is a diagnosis.
21
Constraint Suspension

• Very inefficient
• Scales badly to multiple faults
• Informal

22
Formal Definition of System
23
Use of Abnormal Predicate
24
Syntax of Diagnoses
25
Definition of Diagnosis
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Conflicts
29
From Conflicts to Diagnoses
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A Simple Expository Example
X
6
F
M1
A1
Y
M2
G
A2
Z
M3
31
A Simple Expository Example
X
6
F
M1
A1
6
Y
M2
G
A2
Z
M3
32
A Simple Expository Example
X
6
F
F
M1
A1
12
6
Y
M2
G
A2
Z
M3
33
Derivation of First Minimal Conflict
X
6
10
X
6
X
F
F
F
F
M1
M1
A1
12
A1
6
6
Y
M2
M2
12
G
A2
12
Z
M3
6
34
Derivation of Second Minimal Conflict
X
6
10
X
6
X
F
F
F
F
M1
M1
A1
12
A1
6
6
Y
Y
M2
M2
12
G
G
A2
A2
Z
Z
M3
M3
6
6
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Conflict Directed Search

• Let M be the set of minimal diagnoses, initially
  containing .
• If no more conflicts, the answer is M.
• For every new conflict C
• For every diagnosis D in M
• If D identifies one component in C as faulted, do
  nothing.
• Else remove D from M and add to M all D which
  have some component of C faulted.
• Remove duplicates from M
• Go to 2.

37
Analytical Redundancy Relations
38
Same Diagnoses as Analytical Redundancy

• Assuming
• All ARRs for all observable variables
• Analytic constraints
• Full signature matrix for all multiple faults
• Any deviation indicates a fault
• No fault cancellation

39
Model-Based Diagnosis

• Computes at run-time
• Minimal conflicts and minimal diagnoses usually
  avoid exponential time and space.
• Every system can have a different model.

40
Probabilities

• Assuming components fail independently (p is
  faulted probability) prior probability of a
  diagnosis is
• Bayes Rule

41
Sequential Diagnosis

• Next observation
• Observations are measurements

42
Evaluating
43
Fault M2 low by 2, A2 high by 2
X
6
10
F
F
M1
4
Y
M2
12
G
A2
Z
M3
6

• p0.01
• m16
• Initially p()0.951

44
Measure F10
X
6
10
F
F
M1
4
Y
M2
12
G
A2
Z
M3
6

• Minimal diagnoses A1 M1 M2
• P0.323

45
Measure G12
X
6
10
F
F
M1
4
Y
M2
12
G
A2
Z
M3
6

• P(M1)0.478
• P(A1)0.478
• P(A2,M2).0048

46
Measure X6
X
6
10
F
F
M1
4
Y
M2
12
G
A2
Z
M3
6

• p(A1)0.942
• p(A2,M2)0.0095

• A1 is unfaulted
• The double fault is unlikely

47
Gathering Additional Evidence

• (Assume all measurements are of equal cost)
• Optimal Choose that measurement which, on
  average, yields lowest total diagnosis cost.

Measurements
Outcomes
Measurements
Outcomes
Measurements
Diagnoses
48
Myopic Strategies

• Optimal probing strategies are computationally
  unusable
• Myopic strategies are often close to optimal
• Use one-step lookahead, and use entropy of the
  diagnosis distribution.
• The entropy of a distribution S is

49
Expected Entropy

• Define to be the entropy of the
  diagnosis distribution after measuring
• Can be computed simply by hypothesizing that
• Pick which minimizes

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Fault Modes
55
Modeling Continuous Quantities

• Use techniques from Qualitative Reasoning
• Interval Arithmetic
• Order of Magnitude Reasoning
• Segment each continuous quantity with landmarks
  into distinct regions
• positive, negative, zero
• high, low
• nominal, too high, too low

56
Modeling a Xerographic Copier
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Faster Algorithms

• Minimal conflicts
• Minimal diagnoses
• Myopic probing strategy

BAD NEWS May be exponential number of
conflicts May be exponential number of
diagnoses Not enough to diagnose systems of
10,000s components in any case
59
Compute Diagnoses First!
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Intuitions Underlying Fast Algorithm

• Discover diagnoses with highest prior first
• Only draw inferences which apply to those
  diagnoses
• If conflict free, compute the posterior
  probability
• Continue until sure that the next diagnosis
  discovered will have posterior probability less
  than the ones obtained so far.
• Stop when have the guaranteed n highest posterior
  probability diagnoses

61
cliff criteria met
Best-first search based on prior, avoiding
conflicts
leading diagnosis
CONFLICT
leading diagnosis
leading diagnosis
62
Supervisory Control

• Given
• The commands sent to the system
• The discrete and continuous observations received
  in response
• A model of the system

System Health Management Process
Model
State Estimate
Commands
Observations

• Determine the most likely states of the system to
  enable
• Sensor Validation
• Automatic Reconfiguration/Redundancy Management
• Rapid Repair

63
Three New Ideas

• System evolution over time
• Reconfiguration
• Embedded system

64
Valve Driver Example
Valve Driver
command
Flowv1
Valve1
Pump
Flowv2
Valve2

• Valve driver sends command to valves
• Pump pressurizes the valves
• Flow measured at each valve
• Valve driver may hang, valves may stick shut

65
State Machine Models
Valve2

• Valve driver sends command to valves
• Pump pressurizes the valves
• Flow measured at each valve
• Valve driver may hang, valves may stick shut

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Encoding Device Behavior
Valve Driver Model
cmdInon FailureNone
Off
On
cmdInopen -gt cmdOutopen cmdInclose -gt
cmdOutclose
cmdOut Null
cmdInoff FailureNone
FailureHang
Hung
FailureHang
cmdOut NULL
Prior Probabilities
Value P(FailureValue) None a
Hang 1-a
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Encoding Device Behavior
VDU
cmdInon FailureNone
Off
On
cmdInopen -gt cmdOutopen cmdInclose -gt
cmdOutclose
cmdOut Null
cmdInoff FailureNone
FailureHang
Hung
FailureHang
cmdOut NULL
Turn on Electronics
0
1
68
Encoding Device Behavior
VDU
cmdInon FailureNone
Off
On
cmdInopen -gt cmdOutopen cmdInclose -gt
cmdOutclose
cmdOut Null
cmdInoff FailureNone
FailureHang
Hung
FailureHang
cmdOut NULL
on
off
none
on
null
Turn on Electronics
0
1
69
Encoding Device Behavior
VDU
cmdInon FailureNone
Off
On
cmdInopen -gt cmdOutopen cmdInclose -gt
cmdOutclose
cmdOut Null
cmdInoff FailureNone
FailureHang
Hung
FailureHang
cmdOut NULL
on
off
none
open
on
null
open
Turn on Electronics
0
1
70
Encoding Device Behavior
VDU
cmdInon FailureNone
Off
On
cmdInopen -gt cmdOutopen cmdInclose -gt
cmdOutclose
cmdOut Null
cmdInoff FailureNone
FailureHang
Hung
FailureHang
cmdOut NULL
on
off
none
open
on
null
open
Turn on Electronics
0
1
71
Encoding Device Behavior
VDU
cmdInon FailureNone
Off
On
cmdInopen -gt cmdOutopen cmdInclose -gt
cmdOutclose
cmdOut Null
cmdInoff FailureNone
FailureHang
Hung
FailureHang
cmdOut NULL
Time t
Time t1
on
hung
off
off
VDU
VDU Failure
hang
none
cmdIn
open
on
open
on
null
open
cmdOut
Turn on Electronics
Turn on Electronics
0
1
0
1
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Trajectory Representation
VDU
VDU. Failure
cmdin
cmdout
v1
V1 Failure
V1 Flow
V2
V2 Failure
V2 Flow
4
3
0
1
2
Turn on Electronics
Command valves
Start pump
73
Trajectory Representation
on
off
on
on
VDU
none
none
VDU Failure
none
cmdin
null
on
open
null
cmdout
null
open
closed
open
closed
v1
closed
V1 Failure
V1 Flow
zero
zero
high
zero
open
closed
v2
closed
closed
V2 Failure
V2 Flow
high
zero
zero
zero
3
0
1
2
Turn on Electronics
Command valves
Start pump
74
The Problem with Trajectories
Spacecraft Propulsion System Model

• 10150 states

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Trajectory Representation
on
off
on
on
VDU
none
none
VDU Failure
none
cmdin
null
on
open
null
cmdout
null
open
closed
open
closed
v1
closed
V1 Failure
V1 Flow
zero
zero
high
zero
open
closed
v2
closed
closed
V2 Failure
V2 Flow
high
zero
zero
zero
4
3
0
1
2
Turn on Electronics
Command valves
Observe no flow
Start pump
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Generating Conflicts
on
off
on
VDU
VDU Failure
cmdin
null
on
null
cmdout
null
closed
open
closed
v1
V1 Failure
V1 Flow
zero
zero
high
open
closed
v2
closed
V2 Failure
V2 Flow
high
zero
zero
3
0
1
2
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Supervisory Control
Done

• Given
• The commands sent to the system
• The discrete and continuous observations received
  in response
• A model of the system

System Health Management Process
Model
State Estimate
Commands
Observations

• Determine the most likely states of the system to
  enable
• Sensor Validation
• Automatic Reconfiguration/Redundancy Management
• Rapid Repair

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Choosing Actions
Thrusting
4
3
1
Time
2
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Choosing Actions
Thrusting
4
3
1
Time
2
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Choosing Actions
Tanks Pressurized
Valves Open
Electronics on
Initial State
Thrusting
4
3
0
1
Time
2
Turn on Electronics
Command valves
Ignite Engines
Start pump
81
Choosing Actions
Tanks Pressurized
Valves Open
Electronics on
Initial State
Thrusting
No thrust
Electronics failed
Valves Closed
4
3
0
1
Time
2
Turn on Electronics
Command valves
Ignite Engines
Start pump
82
Choosing Actions
Tanks Pressurized
Valves Open
Electronics on
Initial State
Thrusting
No thrust
Electronics failed
Valves Closed
No thrust, engine ignited with pure fuel (assume
this causes damage)
O2 Valve Sticks
4
3
0
1
Time
2
Turn on Electronics
Command valves
Ignite Engines
Start pump
83
Conflict-based Repair
BlackBox (Kautz Selman, IJCAI 99)
BlackBox is a fast planner for generating
plans from a known initial state
Graphplan/SAT translator
Domain Action Model
SAT Solver
Plan
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Faults in Hybrid Systems Motor Example
Motor does not energize
Fault hypotheses (discrete faults) x (continuous
faults) x (autonomous transition faults)
Motor fault
Motor off
Ramp up
Motor on
Steady state
Motor off
Ramp down
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Hybrid Estimation

• Given continuous and discrete observations,
  determine
• Sequence of modes the system passed through
  (discrete faults)
• Value of continuous parameters (continuous
  faults)
• Time of mode changes (autonomous transition
  faults)

• Challenge Exponential Number of Hypotheses
• Each mode introduces a new set of continuous
  behaviors
• Observations dependent upon entire sequence and
  timing of modes
• Challenge Generally Cannot Factor Hypothesis
  Space
• Failures in autonomous transitions effect when
  each mode was in force
• Changes in continuous parameters may change
  transition timing

86
Particle Filtering for Hybrid Systems

• Challenges
• Hybrid, nonlinear dynamics
• High-resolution diagnosis
• Exponential in number of time steps

• Main ideas
• Focus on a subset of possible modes that cover
  most of the probability space
• Hybrid observer using particle filters with
  automated model switching

87
Current Research Topics in DX

• Titles from the workshop

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Challenges for Model-Based DX

• Noise in observable quantities
• Metric rather than discrete time
• Autonomous transitions
• Continuous degradation
• Modeling of continuous systems (NFIS)
• Thick models
• Prognostics
• Learning

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Challenges for Model-Based DX

• Noise in observable quantities
• Metric rather than discrete time
• Autonomous transitions
• Continuous degradation
• Modeling of continuous systems (NFIS)
• Thick models
• Prognostics
• Learning

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