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4.1 Fundamentals Model-Based Fault Diagnosis Figure 4.1: 3-inverter example This chapter starts by introducing the fundamentals of MBD by means of a classical diagnosis example. Section 4.2 uses this example to introduce the LYDIA approach to MBD. Section 4.3 presents the assumptions of MBD. Section 4.4 discusses issues related to constructing the model of a system. Section 4.5 introduces a metric that can be used to estimate the diagnostic performance of a model-based approach. The final section of this chapter shows how MBD works on a real example. 4.1 Fundamentals MBD is the process of finding differences between behavior predicted by a model, and behavior observed during runtime operation. The model is assumed to be correct, so the differences should be explained by faulty components. Consider Figure 4.1. It shows a simple digital circuit, consisting of 3 inverters: A, B and C. This classical diagnosis example is commonly used to introduce the basic notions of MBD. This text does so likewise. Let w = 1, then y and z should be 1 as well. If observations during runtime indicate that y=0 and z=1, there is a discrepancy between observed and predicted behavior. Any discrepancy is called a symptom, and in this case the symptom is that y=0 while y=1 is predicted. This symptom could be explained by the malfunctioning of inverter B. However, it could also be that both inverter A and inverter C are broken. No subset of {B} or {A, C} is able to explain the symptom, and therefore {B}, {A,C} is called the minimal fault set. Another possible candidate is that all inverters are broken. Actually, all supersets of {B} or {A,C} are candidates, although with lower probability. The question is, how could MBD automate the reasoning of above? The model-based approach requires two artifacts: a model, and a diagnostic engine that operates on that model. These two ingredients of MBD are described now. Model The model of the MBD approach describes the behavior and structure of the components. Let h indicate the health of a component. If h=1, the component is ”healthy” and obeys certain behavior rules. For a combinational system, such as the 3 inverters example, the behavioral rules can be formalized to propositional logic: 34 h A ⇒ (x ⇔ ¬w)
Model-Based Fault Diagnosis 4.1 Fundamentals h B ⇒ (y ⇔ ¬x) h C ⇒ (z ⇔ ¬w) (4.1) This formalization from the concept of 3 inverters to these behavioral rules should be done by humans, and is called the modeling activity. This activity, that could most conveniently be performed by developers of a system, is believed to be the most difficult part of MBD. Diagnostic Engine The second artifact of MBD is a diagnostic engine. This diagnostic engine implements an inference mechanism that is able to produce diagnoses based on any formally described model, and observations made during runtime operation. This can be done by solving the system of equations, as the model provides, by using rules from propositional logic. Figure 4.1 shows the variables that can be observed; w, y and z. Substituting these observables in the system of equations gives: h A ⇒ ¬x h B ⇒ x Then, applying the rule (p ⇒ q) ⇔ (¬p ∨ q) yields: This can be rewritten to DNF-form: h C ⇒ ¬x (4.2) (¬h A ∨ ¬x) ∧ (¬h B ∨ x) ∧ (¬h C ∨ ¬x) (4.3) ¬h A ¬h B ¬h C ∨ ¬h A ¬h B ¬x ∨ ¬h A ¬h C x ∨ ¬h B ¬h C ¬x ∨ ¬h B ¬x = 1 (4.4) Finally, reducing it to the following prime implicants yields: ¬h A ¬h C x ∨ ¬h B ¬x = 1 (4.5) This result is the minimal fault set; either component A and C are broken (if x = 1) or component B is broken (if x = 0). Another way for computing diagnoses is to use conflicts in order to produce the diagnosis. A conflict is a set of components that cannot be healthy all together. For example, given the symptom ”y=0 while y=1 is predicted”, the set {A, B, C} is a conflict. In other words, the assumption that component A, B and C are all healthy should be removed. Section 3.5 used this view on solving the diagnosis problem, for giving the reader a first idea on MBD. A minimal conflict is a set of components that is no longer a conflict if you remove one of its members. These are the interesting sets, because they correspond to diagnoses with high probability. Applying the resolution rule (p ∨ q) ∧ (r ∨ ¬q) ⇒ (p ∨ r) to Equation 4.3 yields: Then, using De Morgan’s Laws: (¬h A ∨ ¬h B ) ∧ (¬h B ∨ ¬h C ) = 1 (4.6) ¬( (¬h A ∨ ¬h B ) ∧ (¬h B ∨ ¬h C ) ) = 0 ¬(¬h A ∨ ¬h B ) ∨ ¬(¬h B ∨ ¬h C ) = 0 ¬h A ¬h B ∨ ¬h B ¬h C = 0 (4.7) 35
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Page 4 and 5: © 2006 W.M. Lindhoud. Document gen
Page 7: Preface The work presented in this
Page 10 and 11: CONTENTS 4.4 Modeling . . . . . . .
Page 13: List of Tables 3.1 Evaluation appro
Page 16 and 17: 1.1 Fault Diagnosis Introduction Fi
Page 18 and 19: 1.2 Automated Fault Diagnosis Intro
Page 20 and 21: 1.5 Outline of the Thesis Introduct
Page 23 and 24: Chapter 2 State-of-the-Practice Fau
Page 25 and 26: State-of-the-Practice Fault diagnos
Page 33 and 34: Chapter 3 Automated Fault Diagnosis
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Page 41 and 42: Automated Fault Diagnosis 3.3 Off-l
Page 43 and 44: Automated Fault Diagnosis 3.5 Model
Page 45: Automated Fault Diagnosis 3.6 Evalu
Page 50 and 51: 4.2 Model-Based Diagnosis with LYDI
Page 52 and 53: 4.3 Basic Assumptions Model-Based F
Page 54 and 55: 4.4 Modeling Model-Based Fault Diag
Page 56 and 57: 4.4 Modeling Model-Based Fault Diag
Page 58 and 59: 4.5 Entropy Gain Model-Based Fault
Page 60 and 61: 4.6 MBD on the Power Supply Model-B
Page 63 and 64: Chapter 5 Diagnosing the Beam Prope
Page 65 and 66: Diagnosing the Beam Propeller Movem
Page 83 and 84: Chapter 6 Conclusions and Recommend
Page 85: Conclusions and Recommendations 6.2
Page 88 and 89: BIBLIOGRAPHY [12] Johan de Kleer an
Page 91 and 92: Appendix A Glossary of terms In thi
Page 93 and 94: Glossary of terms • Diagnostic Pe
Page 95 and 96: Glossary of terms • Model: an abs
Page 97: Glossary of terms • System: an en
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B.2 System Data Case Study Details
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B.4 Full List of Examined Fault Sce
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B.5 Discretization of Implementatio
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C.1 The Synthetic Models that are U
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C.2 Model of the Power Supply Examp
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C.3 The Models Constructed for the
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Appendix D State-of-the-Future Faul
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