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4.5 Entropy Gain Model-Based Fault Diagnosis Figure 4.2: 4-inverter example are already known; the input x is true and the output y is false. The considered points are referred to by a, b and c. The prerequisite for the entropy calculation is the assignment of an a priori probability to each health variable. This a priori probability is the probability that h=0, without having made any observations. The 4-inverter system has 4 health variables, with a priori probabilities: p(¬h A ) = 0.01 p(¬h B ) = 0.01 p(¬h C ) = 0.01 p(¬h D ) = 0.01 (4.8) The interpretation of, for example p(¬h A ) = 0.01, is not that randomly picking n components yields 0.01 ∗ n broken components. The meaning depends on the model of the system. If no observations have been made, all single faults and multiple faults are possible. There are 4 health variables. This means that if x and y are not observed, the number of possible diagnoses is 2 4 = 16. This information, the existence of 16 possible diagnoses, can be stored in 4 bits. The corresponding outcome of the LYDIA diagnostic engine is as follows: (0.960596) hA = true, hB = true, hC = true, hD = true (0.00970299) hA = false, hB = true, hC = true, hD = true (0.00970299) hA = true, hB = false, hC = true, hD = true (0.00970299) hA = true, hB = true, hC = false, hD = true (0.00970299) hA = true, hB = true, hC = true, hD = false ... The firstly listed diagnosis, that states that all inverters are healthy, has highest probability. Specific diagnoses are referred to as health vectors. The correct functioning of all inverters is denoted by the health vector ¯h = {1,1,1,1}, and its probability is calculated by using the formula: p({h A ,h B ,h C ,h D }) = p(hA) ∗ p(hB) ∗ p(hC) ∗ p(hD) (4.9) This formula assumes that the variables are independent: the correct or incorrect functioning of one component has no effect on the health of any other component. The interpretation of, p(¯h = {1,1,1,1}) = 0.970596, is that diagnosing the 4-inverter system at an arbitrary moment, without any observations being made until that moment, yields a probability of 0.97% that the entire system is healthy. p(¯h = {0,1,1,1}) = 0.00970299 means that there is a probability of 0.01% that hA=0. Also, there is probability of 0.01% that hB=0, etcetera. The a priori probabilities of the health vectors allow for the calculation of entropy. The entropy (H) is defined as: H = −∑ p i log p i , (4.10) 44
Model-Based Fault Diagnosis 4.6 MBD on the Power Supply Nr. x hA a hB b hC c hD y p 1 1 0 1 0 1 0 1 1 0 0.00000099 2 1 0 1 0 1 1 0 0 0 0.00000099 3 1 0 1 1 0 0 0 0 0 0.00000099 4 1 0 1 1 0 1 1 1 0 0.00970299 5 1 1 0 0 0 0 0 0 0 0.00000099 6 1 1 0 0 0 1 1 1 0 0.00970299 7 1 1 0 1 1 0 1 1 0 0.00970299 8 1 1 0 1 1 1 0 0 0 0.00970299 Table 4.1: Mode catalog of the 4-inverter model, with x=1 and y=0. where p i is the probability that health vector i is the actual candidate given a specific set of observations. Formula 4.10 is a cost function to estimate the expected cost of identifying the actual candidate. It is constructed as follows. The cost of locating a particular candidate is proportional to log p −1 i (so, a binary search through p −1 i objects). The expected costs of identifying one candidate is the multiplication of the costs to locate it (log p −1 i ) and the probability that the candidate is the actual candidate (p i ). The entropy H, defined by Formula 4.10, adds the costs of all candidates (∑ p i log p −1 i = −∑ p i log p i ). A mode catalog is a table that specifies all possible observations for each health vector, and can be derived from the model. The used 4-inverter model, as listed in Appendix C.1.4, is a strong model, and the fault mode of 1 inverter is defined by: ¬h => (o = i). The mode catalog of this 4-inverter model contains 8 entries, when none of the observables a, b, and c are observed. This information, the existence of 8 health vectors, can be stored in 3 bits, in contrast to the 4 bits needed for storing the information that there are 16 health vectors. This does not imply that the entropy gain is 4 − 3 = 1 bit, because entropy also depends on the probability of each health candidate (the entropy gain of measuring the input x and output y is 0.064). Table 4.1 shows the 4-inverter model, when x=1 and y=0. The entropy of this mode catalog (no additional observation are made), is 0.260. This means that the uncertainty of the outcome of the MBD engine is 0.260. Consider that a is being measured. There are two possibilities: a=0 or a=1. In either case another set of entries is consistent with the observations. These possible new mode catalogs define the cost to locate the actual candidate. The expected entropy of observing a is 0.046. The entropy gain is defined as the entropy a priori model minus the expected entropy of observing a: 0.260 − 0.046 = 0.213. The entropy gain of observing b and c is respectively 0.221 and 0.213. Thus, the entropy gain of observing b is highest, and is the best choice for to measure. 4.6 MBD on the Power Supply This section presents a real-life application of MBD on the power supply example. The model is shown in Appendix C.2.1. The model specifies six systems, namely Cable, Fuse, Low_ voltage_ power_ supply, PDU, Unit, and Power_ Supply. The specification of the behavior is very abstract. The model describes the power supply example as a digital system. The used variables are all in the boolean domain. A value of 1 denotes that there is current, and a value of 0 denotes that there is no current. For example, the model of a Cable specifies that when a cable is healthy it conducts current, using the following definition: 45
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Automated Fault Diagnosis at Philip
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© 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
Page 35 and 36: Automated Fault Diagnosis 3.1 Overv
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 48 and 49: 4.1 Fundamentals Model-Based Fault
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 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
Page 100 and 101: B.2 System Data Case Study Details
Page 102 and 103: B.4 Full List of Examined Fault Sce
Page 104 and 105: B.5 Discretization of Implementatio
Page 106 and 107: 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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