knowledge-based systems for industrial...

Model-Based Systems & Qualitative Reasoning

Group of the Technical University of Munich SS 15 KBSIA 4 - 17

Knowledge-based Systems for Industrial Applications

1 The Topic

2 Tasks

3 Modeling

4 Diagnosis

4.2 Component-oriented

Diagnosis

Goal:

Restriction to special classes of

systems, fault types, problem

classes

Algorithms for different diagnosis

tasks

Script: Chap. 10.4.1, 10.4.2, 10.4.3



Component-oriented Diagnosis

Components:

• (Usually) physical objects

• interacting in a fixed structure

• of an artifact

• i.e. a (well-)designed system

Assumptions:

• System:

components + structure

• System behavior:

component behavior + structure

(reductionism)

• Correct system behavior achieves

goals

(well-designed system)

• Correct component behavior

achieves correct system behavior

• Diagnostic assumption:

no undesigned components or

interactions


Group of the Technical University of Munich - 19 SS 15 KBSIA 4

Component-oriented Diagnosis - Examples

• Electrical Circuits

• Hydraulic Circuits

• Pneumatic Systems

• Mechatronic Systems

• ...

30

V1

Ignition 15

(22)K

15-E

(39)E

KP

-0

(1)B

AT

+

(19)B

AT

+

(33)H

RL-0

(4)B

AP

+

Control Unit

BA

T-(

16)

BA

T-(

52)

V2

31

MEKP

2

31

5

45

EKP

Relay

System Relay

Main relay

hydraulic unit

front left wheel

rear right wheel

brake pedal



Different Diagnosis Tasks

• Is the system working correctly?

Has a fault occurred?

• In which component has a fault

occurred?

• Which fault has occurred?

• Monitoring Fault detection

• Fault localization

• Fault identification



Behavior Modes

• System comprises a (finite) set of components

COMPS = { Ci }

• Each Ci has a (finite) set of behavior modes

modes(Ci) = { mij(Ci)}

• E.g.

- (unique) correct behavior: ok(Ci)

- (any) faulty behavior: ok(Ci)

- a specific fault: stuck-closed(valvei)

• Behavior mode operating mode

(of correct behavior)

• E.g. blocking mode of a diode

Definition (Mode Assignment)

• COMPS’ COMPS

• MA = {mij(Ci) Ci COMPS’ }

• or MA = Ci COMPS’ mij(Ci)

• MA complete: COMPS’=COMPS


Group of the Technical University of Munich SS 15 KBSIA 4

Behavior Models

• System behavior:

component behavior

+ structure

• System model:

component models

+ structure description

• Library associates a (relational)

behavior model with each

behavior mode

• System model:

library + mode assignment

+ structure description

• mij(Ci) modelij

(Ci)

• MODEL =

LIB {mij(Ci) Ci COMPS }

STRUCTURE

• In [Reiter 87]:

SD = LIB STRUCTURE

• MODEL = SD MA

- 22



Detection of Component Faults

• Has a fault occurred?

Is the system working correctly?

• Do observations provide evidence

that the system working according

to its goal specification?

• Assumption: Correct system

behavior achieves goals

• Are observations consistent

with system model of correct

behavior?

MODELOK ⊨ GOALS

Fault Detection:

SD MAok OBS ⊨ ? ^

• MAok = {mok (Ci) Ci COMPS }

• MODELOK = SD MAok

MODELok OBS ⊨? ^


Group of the Technical University of Munich

The Stupid System – Detecting Inconsistencies

• Inconsistent partial models:

„conflict“

- 24 SS 15 KBSIA 4



Knowledge-based Systems for Industrial Applications

1 The Topic

2 Tasks

3 Modeling

4 Diagnosis

4.2 Component-oriented

Diagnosis

4.2.1 Fault localization

Goal:

Definition

Characterization

Computation

Script: Chap. 10.4.1, 10.4.2



Diagnosis - Definition

• Diagnosis (hypotheses):

Models that are consistent with observations

Definition (Diagnosis):

• A complete mode assignment MA that is consistent with the

observations:

SD MA OBS ⊭ ^

Definition (Abductive diagnosis):

• A complete mode assignment MA that entails the observations:

SD MA ⊨ OBS



Fault Localization - Definition

• For each component:

• Ok or not ok?

• modes(Ci) = { oki(Ci), oki(Ci) }

• All modes different from ok imply ok:

• mode(C) modes(C) \ {ok(C)}

mode(C) ok(C)

Definition (Fault Localization)

• OK COMPS

• FAULTY = COMPS \ OK is a fault localization iff

• MA(FAULTY) := Ci FAULTY oki(Ci) Ci OK oki(Ci)

is a diagnosis of SD and OBS

SD MA(FAULTY) OBS ⊭ ^



Fault Localization - Example

M1

M2

M3

A1

A2

F = 10

G = 12

2

3

2 3

3

3 Multipliers

and 2 Adder

List all fault

localizations!

• {A1}

• { M1}

• {A2, M2},

• {M2, M3}

• {A1, M1}, ...

• {A1, M2, M3}, ...

• {A1, M2, M3 , A2}, ...

• {A1, A2, M1, M2, M3} !!



Minimal Fault Localization

• Why assume more components to be broken than necessary?

• “Occam’s razor”

Definition (Minimal Fault Localization)

• FAULTY is a minimal fault localization iff

• no proper subset FAULTY’ FAULTY is a fault localization



Fault Localization - Example

• {A1, M1}, ...

• {A1, M2, M3}, ...

• {A1, M2, M3 , A2}, ...

• {A1, A2, M1, M2, M3} !!

List all fault

localizations!

• {A1}

• { M1}

• {A2, M2},

• {M2, M3}

M1

M2

M3

A1

A2

F = 10

G = 12

2

3

2 3

3

3 Multipliers

and 2 Adder

List all minimal fault

localizations!



If all components in {A1, M2, M1} were ok, F=12 would hold

Contradiction!

One of them must be broken

Partial inconsistent mode assignments: important diagnosis

information

How to Find Fault Localizations? - Example

M1

M2

M3

A1

A2

F = 10

G = 12

2

3

2 3

3

3 Multipliers

and 2 Adder



How to Find Fault Localizations? - Conflicts

Definition (Conflict)

• If the mode assignment MA = Ci COMPS’ mij(Ci)

is inconsistent with the observations:

SD MA OBS ⊨ ^

• then its negation Ci COMPS’ mij(Ci) is called a conflict

M1

M2

M3

A1

A2

F = 10

G = 12

2

3

2 3

3

3 Multipliers

and 2 Adder



The Stupid System – Conflicts

• Inconsistent partial models:

„conflict“

- 33 SS 15 KBSIA 4


Group of the Technical University of Munich - 34

hydraulic unit

front left wheel

rear right wheel

brake pedal

under-

braked

over-

braked

harder

Conflicts - Example ABS Hydraulics

• Conflict:

ok(left inlet valve)

ok(left outlet valve)

• Provided the pedal, wheel, pipes etc. are OK

hydraulic unit

front left wheel

rear right wheel

brake pedal

under-

braked

over-

braked

harder

Discrepancy!

{LOV}: [QLOV] = 0

[pWBC] = [–]

[pWBC] = [–]

[pMC] = [+]

[QLIV] [QLOV] = [–]

{LIV}: [QLIV] = [+]

{LOV}: [QLIV] = [–]

Under-braked

harder

SS 15 KBSIA 4


Group of the Technical University of Munich SS 15 KBSIA 4 - 35 - 35

ABS Hydraulics: Second Conflict

{LIV, ROV, RIV, THR}: [QPMP] = [+]{PMP}: [QPMP] = [0]

[QPED] [QLIV] [QTHR] [QRIV] = [0]

discrepancy!

[QPED] = [–]

{LIV, ROV, RIV, THR}: [pDC] = [+]

{LIV, ROV, RIV}: [QTHR] = [–]

{LIV}: [QLIV] = [+] {RIV, ROV}: [QRIV] = [+]

{LOV}: [QLIV] = [–]

under-braked over-braked

harder

• Conflict:



• Conflict #2:


ok(right inlet valve)

ok(right outlet valve)

ok(throttle)

ok( pump)



Minimal Conflicts - All You Need for Diagnosis

• Each restriction to less modes corresponds to a consistent

mode assignment

• “sharpest” characterization of possible diagnoses

• Even more: they capture the entire diagnosis information

Theorem

• A mode assignment MA is a diagnosis of SD OBS

• iff it is consistent with the minimal conflicts of SD OBS

-



Minimal Conflicts Replace SD OBS

Theorem

• A mode assignment MA is a diagnosis of SD OBS

• iff it is consistent with the minimal conflicts of SD OBS

SD OBS MA ⊭ ^

MIN-CONFL MA ⊭ ^

Not needed any more:

• Model

• Observations

• Predictions

• Discrepancies and

• their magnitude ...



Positive Conflicts - All You Need for Fault Localization

• Positive conflicts:

Ci COMPS’ ok(Ci)

• at least one of the components mentioned is broken

• minimal positive conflicts:

“sharpest” characterization of fault locations

• If there are no fault models, fault modes make no predictions

• all conflicts are positive

Theorem

• FAULTY COMPS is a minimal fault localization

• iff Ci FAULTY ok(Ci)

is a prime implicant of the positive minimal conflicts



Prime Implicants

Definition (Implicant)

• Th: a set of propositional formulas

• CC: conjunction clause (of literals)

• CC is an implicant of Th

• iff it entails all formulas in Th

Theorem


• iff Ci FAULTY ok(Ci)

is a prime implicant of the positive minimal conflicts

CC ⊨ Th

• CC is a prime implicant of Th

• iff no proper sub clause CC’

is also an implicant of Th

CC’ ⊭ Th



Prime Implicants - Example

Definition (Implicant)

• Th: a set of propositional formulas

• CC: conjunction clause (of literals)

• CC is an implicant of Th

• iff it entails all formulas in Th

CC ⊨ Th

• CC is a prime implicant of Th

• iff no proper sub clause CC’

is also an implicant of Th

CC’ ⊭ Th

• Th:

{ A B ,

A C D}

• Some implicants:

A

B C

A C

A B C D

B D

• Prime implicants:

A

B C

B D



The Stupid System – Fault Localizations

Conflicts:

ok(flow-sensor)

ok(pump) ok(container) ok(pressure-sensor)

• ok(mech-drive)

ok(pump) ok(container) ok(pressure-sensor)

Minimal fault localizations:

ok(pump)

ok(container)

ok(pressure-sensor)

• ok(mech-drive) ok(flow-sensor)

SS 15 KBSIA 4 - 41



ABS Hydraulics: Fault Localizations

• Conflict:



• {left inlet valve}

• {left outlet valve,

right inlet valve}


right outlet valve}


throttle}


pump}

• Conflict #2:




ok(throttle)

ok( pump)



• {left inlet valve}


right inlet valve}


right outlet valve}


throttle}


pump}

Computing Fault Localizations - Idea

• Conflict:



• Each fault localization has to contain at least one component out of each positive conflict

• Conflict #2:




ok(throttle)

ok( pump)



Computing Fault Localizations as Hitting Sets

Definition (Hitting Set)

• M = { Mi } a set of sets

• HS is called a hitting set of M

• iff HS contains at least one element

out of each Mi :

• Mi M HS Mi

• HS is a minimal hitting set of M, if

no proper subset of HS is one

Theorem


• iff it is a minimal hitting set of the set of the sets of

components that occur in the positive minimal conflicts



Min. hitting sets of {{A1, M2, M1}}:

{A1}, { M2}, {M1}

ok(A1) ok(M2) ok(M1) F=12

ok(A1) ok(M2) ok(M1) F=10 inconsistent

minimal conflict: ok(A1) ok(M2) ok(M1)

Computing Fault Localizations - Adder-Multiplier Example

BUT:

Minimal fault

localizations:

• {A1}

• { M1}

• {A2, M2},

• {M2, M3} ?

M1

M2

M3

A1

A2

F = 10

G = 12

2

3

2 3

3

3 Multipliers

and 2 Adder



Min. hitting sets of {{A1, M2, M1}}:

{A1}, { M2}, {M1}

Min. hitting sets of {{A1, M2, M1}, {M1, M3, A1, A2}}:

{A1}, {M1}, {A2, M2}, {M2, M3}

ok(M1) ok(A1) ok(M3) ok(A2) G=10

ok(M1) ok(A1) ok(M3) ok(A2) G=12 incons.

conflict: ok(M1) ok(A1) ok(M3) ok(A2)

Adder-Multiplier Example - The Second Conflict

BUT:

Minimal fault

localizations:

• {A1}

• { M1}

• {A2, M2},

• {M2, M3} !

M1

M2

M3

A1

A2

F = 10

G = 12

2

3

2 3

3 6

6 3 Multipliers

and 2 Adder

4

10



Fault Localization - The Overall Picture

Model Revision

Conflict

Fault Localization

No fault models

no consistency check

Discrepancy

Diagnosis

Predictor

Observations/Goals

Behavior Model

Library

ok Modes

Model Composer

E.g. Hitting Sets



But: How to Obtain Conflicts?

Model Revision

Conflict

Fault Localization

Discrepancy

Diagnosis

Predictor

Observations/Goals

Behavior Model

Library

ok Modes

Model Composer ?



Dependency-based diagnosis - The Idea

ok(M1) ok(A1) ok(M3) ok(A2) G=10

ok(M1) ok(A1) ok(M3) ok(A2) G=12 incons.

conflict: ok(M1) ok(A1) ok(M3) ok(A2)

M1

M2

M3

A1

A2

F = 10

G = 12

2

3

2 3

3 6

6 3 Multipliers

and 2 Adder

4

10

Record dependencies:

• of constraints on modes

• of values on applied constraints

• of discrepancies on values



Dependencies

Model

composition { Behavior

Modes M1 M4 M3 M2

C4 C1 C2 C3 Constraints

P3 P1 P2 Predictions

Discrepancies

{ Behavior Prediction

{ Discrepancy

Detection



Recording Dependencies

Behavior

Modes M1 M4 M3 M2

C4 C1 C2 C3 Constraints

P3 P1 P2 Predictions

Discrepancies

Model

composition { { Behavior Prediction

{ Discrepancy

Detection

{M1} {M4} {M3} {M2}

{M1} {M1 M2}

{M3

M4}

{M1 M2 M3

M1 M2 M4}



ATMS: Basic Concepts

General Tool:

Assumption-based Truth-

Maintenance System

ATMS node

- (Elementary) proposition

Assumption

- Truth of a proposition

Justification

- Elementary inference

E.g. P1 C2 P2

M1 M4 M3 M2

C4 C1 C2 C3

P3 P1 P2

{M1} {M4} {M3} {M2}

{M1} {M1 M2}

{M3 }

{M4}

{M1 M2 M3

M1 M2 M4}

M1



ATMS: Recording Dependencies in Labels

Environment:

- (Consistent) set

(conjunction) of assumptions

deriving a node

E.g. M1 M2

Label:

- Set (disjunction) of

environments that are minimal

w.r.t. set inclusion

Nogood:

- Inconsistent set

(conjunction) of assumptions

M1 M4 M3 M2

C4 C1 C2 C3

P3 P1 P2

{M1} {M4} {M3} {M2}

{M1} {M1 M2}

{M3 }

{M4}

{M1 M2 M3

M1 M2 M4}

M1



Task of the ATMS

Computes and maintains

labels:

- when justifications and

assumptions are added

- removes inconsistent

environments

- minimizes environments

Computes (minimal) nogoods

M1 M4 M3 M2

C4 C1 C2 C3

P3 P1 P2

{M1} {M4} {M3} {M2}

{M1} {M1 M2}

{M3

M4}

{M1 M2 M3

M1 M2 M4}

M1

For diagnosis:

- computes minimal conflicts

(= negated minimal inconsistent mode assignments)

(= negated minimal nogoods)



Conflict Generation Using an ATMS

Model Revision

Conflict

Fault Localization

Discrepancy

Diagnosis

Predictor

Observations/Goals

Behavior Model

Library

ok Modes

Model Composer ? Dependencies

ATMS

Conflict Generator

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