Diagnosing Multiple Faults with the Dynamic Binary Matrix

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Diagnosing Multiple Faults with Diagnosing Multiple Faults with Diagnosing Multiple Faults with Dynamic Binary Matrix Dynamic Binary Matrix Dynamic Binary Matrix

the the the

Michal Barty´ s Micha ss Michalll Barty´ Barty´ Micha Barty´ s Institute of Automatic Control and Robotics, Warsaw University of Institute Automatic Control and Robotics, Warsaw University of Institute of of´sw. Automatic Control and Warsaw, Robotics,Poland Warsaw University of Technology, A. Boboli 8, 02-525 (e-mail: bartys@ Institute of Automatic Control and Robotics, Warsaw University of Technology, ´s´sw. A. Boboli 8, 02-525 Warsaw, Poland (e-mail: bartys@ Technology, w. A. Boboli 8, 02-525 Warsaw, Poland (e-mail: bartys@ mchtr.pw.edu.pl) Technology, ´sw. A. Boboli 8, 02-525 Warsaw, Poland (e-mail: bartys@ mchtr.pw.edu.pl) mchtr.pw.edu.pl) mchtr.pw.edu.pl) Abstract: This paper presents the underlying theory of diagnosing multiple faults with the Abstract: This paper the underlying theory of multiple faults the Abstract: This diagnostic paper presents presents theThe underlying theorystatements of diagnosing diagnosing multiple faults with with and the binary dynamic matrix. fundamental regarding inconsistency Abstract: This paper presents the underlying theory of diagnosing multiple faults with the binary dynamic diagnostic matrix. The fundamental statements regarding inconsistency and binary dynamic diagnostic matrix. The fundamental statements regarding inconsistency and multiple fault isolation with matrix. the dynamic diagnostic matricesregarding are formulated and proved. binary dynamic diagnostic The binary fundamental statements inconsistency and multiple fault isolation the binary diagnostic matrices multiplean fault isolationofwith with the dynamic dynamic binary is diagnostic matrices are are formulated formulated and and proved. proved. Finally, algorithm multiple fault isolation presented. multiple fault isolation with the dynamic binary diagnostic matrices are formulated and proved. Finally, an algorithm of multiple fault isolation is presented. Finally, an algorithm of multiple fault isolation is presented. Finally, an algorithm of multiple fault isolation is presented. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: dynamic binary diagnostic matrix, multiple fault isolation, fault diagnosis, Keywords: dynamic binary diagnostic matrix, multiple fault fault diagnosis, Keywords: dynamic dynamic binarydiagnostic diagnosticmatrix, matrix,diagnostic multiple fault fault isolation, isolation, faultfault diagnosis, inconsistency of dynamic inference, multiple isolation Keywords: binary diagnostic matrix, multiple isolation, fault diagnosis, inconsistency of dynamic diagnostic matrix, diagnostic inference, multiple fault isolation inconsistency of dynamic diagnostic matrix, diagnostic inference, multiple fault isolation algorithm. inconsistency of dynamic diagnostic matrix, diagnostic inference, multiple fault isolation algorithm. algorithm. algorithm. d 1. INTRODUCTION nostic Matrix (BDM dd ). Therefore, this paper refers to 1. INTRODUCTION d). Therefore, this paper refers to nostic Matrix (BDM 1. INTRODUCTION nostic Matrix (BDM ). Therefore,of this this paperDetection refers to to fault location from the perspective the Fault 1. INTRODUCTION nostic Matrix (BDM d ). Therefore, paper refers location from the perspective of the Fault Detection The multiple faults are considered in this paper as the fault fault location from the perspective of the Fault Detection and Isolation (F DI) rather then from Artificial Intellocation from the perspective of the Fault Detection The faults this paper as The multiple multiple faults are are considered considered in thisfaults. paperMostly, as the the fault and (F rather then Artificial Intelaggregates of simultaneously existingin single and Isolation Isolation (F DI) DI) despite rather promising then from from results Artificial IntelThe multiple faults are considered in this paper as the ligence methodology achieved and Isolation (F DI) rather then from Artificial Intelaggregates of simultaneously existing single faults. Mostly, aggregates of simultaneously existing single faults. Mostly, ligence methodology despite promising results achieved the signature of the multiple fault is assumed as a logical ligence methodology despite promising results achieved aggregates of simultaneously existing single faults. Mostly, ligence recently by Nyberg (2011). He proposed the algorithm methodology despite promising results achieved the the fault is as logical the signature signature ofthese the multiple multiple fault is assumed assumed as aaaKorbicz logical providing recently (2011). He the alternative of of signatures (Gertler (1998); recently by bya Nyberg Nyberg (2011). He proposed proposed the algorithm algorithm the signature of the multiple fault is assumed as logical significant performance improvement comrecently by Nyberg (2011). He proposed the algorithm alternative of these signatures (Gertler (1998); Korbicz alternative of these signatures (Gertler (1998); Korbicz aaDI significant performance improvement comet al. (2004); Blanke et al. (2006); Isermann (2006); Pat- providing providing significant performance improvement comalternative of these signatures (Gertler (1998); Korbicz pared to F approaches based on structured residuproviding a significant performance improvement comet al. (2004); Blanke et al. (2006); Isermann (2006); Patet al. (2004); Blanke et al. (2006); Isermann (2006); Patpared to F DI approaches based on structured residuton et al. (2000); Nyberg (2006); Trav´ e -Massuy´ e s (2014)). pared to F F DI DItheapproaches approaches based on onwithin structured residuet al. (2004); Blanke et al. (2006); Isermann (2006); Pat- pared als. Anyway, last achievements F DI, based to based structured residuton Nyberg Trav´ ee-Massuy´ eess (2014)). ton et et al. al. (2000); (2000); Nyberg (2006); (2006); Trav´ -Massuy´ (2014)). als. Anyway, the achievements F based Therefore, the hypothesis regarding multiple faults is credals. dynamic Anyway, binary the last last achievements within F DI, DI,(Barty´ based ton et al. (2000); Nyberg (2006); Trav´ e-Massuy´ es (2014)). on diagnostic matrixwithin approach s als. Anyway, the last achievements within F DI, based Therefore, the hypothesis regarding multiple faults is Therefore, the hypothesis regarding multiple faults is credcreddynamic binary diagnostic matrix approach (Barty´ ss ible as longthe ashypothesis logically combined of the cur- on on dynamic binary diagnostic matrix approach (Barty´ Therefore, regarding signatures multiple faults is cred(2014a,b); Ko´ s cielny et al. (2012)) still remain competitive. on dynamic binary diagnostic matrix approach (Barty´ s ible as long as logically combined signatures of the curible as long as logically combined signatures of the cur(2014a,b); Ko´ s cielny et al. (2012)) still remain competitive. rently existing single faults match a vector of binary eval(2014a,b); Ko´ scielny cielny et etbinary al. (2012)) (2012)) still remain remain competitive. ible as long as logically combined signatures of the cur- The idea ofKo´ a dynamic diagnostic matrixcompetitive. itself is not (2014a,b); s al. still rently existing single faults match a vector of binary evalrently values existing single faultstests. match vector of of binary binary eval- The of dynamic binary diagnostic matrix uated of single diagnostic The regarding The idea of a dynamic binary diagnostic matrix itself is is not not rently existing faults match aa assumption vector evalto beidea considered as novel. Primary, it has been itself introduced The idea of aa dynamic binary diagnostic matrix itself is not uated tests. The assumption uated values values of of diagnostic tests. Thesignatures assumptionasregarding regarding to be considered as novel. Primary, it has been introduced composition of diagnostic the multiple fault alterna- by to be considered as novel. Primary, it has been introduced uated values of diagnostic tests. The assumption regarding Ko´ s cielny (1995) for the purposes of the decomposito be considered as novel. Primary, it has been introduced composition the fault alternacomposition offaults the multiple multiple fault signatures signatures as alterna(1995) the of tives of singleof has a serious drawback. as There is a by by Ko´ Ko´ cielny (1995) for formatrix the purposes purposes of the the decomposidecomposicomposition of the multiple fault signatures as alternation ofssscielny the diagnostic in the Dynamic Table of Ko´ cielny (1995) for the purposes of the decompositives of single faults has aa multiple serious drawback. There is aa by tives of single faults has serious drawback. There is tion of the diagnostic matrix in the Dynamic Table finite probability, that the faults may cause the tion of the diagnostic matrix in the Dynamic Table of of tives of single faults has a serious drawback. There is a States approach (DT S). The decomposition of diagnosed tion of the diagnostic matrix in the Dynamic Table of finite probability, that the multiple faults may cause the finite probability, that the multiple faults may cause the States approach (DT S). The decomposition of diagnosed effect referred to as residual cancellation phenomenon; see States approach (DT S). The decomposition of diagnosed finite probability, that the multiple faults may cause the States system in DT S approach relies on a dynamic creation of approach (DT S). The decomposition of diagnosed effect as cancellation phenomenon; effect referred referred toThis as residual residual cancellation phenomenon; see in approach relies on dynamic creation Gertler (1998).to phenomenon relies on cancellationsee of system system in DT DT S S fault approach reliessubstructures on a dynamic creation of of effect referred to as residual cancellation phenomenon; see the appropriate isolation (subsystems) in DT S approach relies on aa dynamic creation of Gertler (1998). This phenomenon relies on cancellation of Gertler (1998). This phenomenon relies on cancellation of system the appropriate fault isolation substructures (subsystems) some residuals incidental dynamic static mutual the appropriate fault isolation substructures (subsystems) Gertler (1998). due Thistophenomenon relies onorcancellation of that consist exclusively of all possible faults and diagnostic appropriate fault isolation substructures (subsystems) some due incidental or mutual some residuals residuals due to tovalues incidental dynamic or static static mutual the that exclusively of possible faults diagnostic inference of peculiar of thedynamic single faults. Therefore, that consist consist exclusively of all all possible faults and and diagnostic some residuals due to incidental dynamic or static mutual signals sensitive to these faults depending on the actual that consist exclusively of all possible faults and diagnostic inference of peculiar values of the single faults. Therefore, inference of peculiar values of the single faults. Therefore, signals sensitive to these faults depending on the actual based on structures the isolation of exclusively single faults signals sensitive to these faults depending on the actual inference of peculiar values of the single faults. Therefore, signals (dynamic) results of diagnostic tests. The new subsystem sensitive to these faults depending on the actual the isolation of single faults on theresidual isolationsets of exclusively exclusively singlethe faults based on structures structures (dynamic) results of diagnostic tests. The new subsystem of cannot exclude casebased of multiple faults is (dynamic) results of diagnostic tests. The new subsystem the isolation of exclusively single faults based on structures created immediately just after appearance of any new results of diagnostic tests. The new subsystem of sets exclude case of faults of residual residual sets cannot exclude the caseexclude of multiple multiple faults (dynamic) is created immediately just after appearance of any new and isolation ofcannot multiple faults the do not the possiis created immediately just after appearance of any new of residual sets cannot exclude the case of multiple faults fault symptom. Each time a new dynamic subsystem is is created immediately just after appearance of any new and isolation of multiple faults do not exclude the possiand isolation of multiple faults do not exclude the possifault symptom. Each time a new dynamic subsystem is bility of the existence of single faults vice versa. Hence, fault symptom. Each time a new dynamic subsystem is and isolation of multiple faults do not exclude the possi- created, the set of actual symptoms are checked for infault symptom. Each time a new dynamic subsystem is bility of the existence of single faults vice versa. Hence, bility of the existence of single faults vice versa. Hence, created, the set of actual symptoms are checked for inin general, the problem of multiple fault isolation is not created, the set of actual symptoms are checked for inbility of the existence of single faults vice versa. Hence, created, consistency within the static Binary Diagnostic Matrix the set of actual symptoms are checked for ingeneral, the of fault is intrivial general, theInproblem problem of multiple multiple fault isolation isolation is not not consistency within static Diagnostic Matrix ain task. this paper we will assume that residual consistency within the static Binary Diagnostic Matrix in general, the problem of multiple fault isolation is not (BDM ). Ko´ scielny the defined theBinary problem of inconsistency consistency within the static Binary Diagnostic Matrix aa trivial task. In this paper we will assume that residual trivial task. In this paper we will assume that residual ). Ko´ ssapproach. cielny defined the problem of inconsistency cancellation phenomena do not take place because of their (BDM ). Ko´ cielny defined the problem of inconsistency a trivial task. In this paper we will assume that residual (BDM in the DT S He assumed inconsistency as the ). Ko´scielny defined the problem of inconsistency cancellation phenomena do cancellation phenomena do not not take take place place because because of of their their (BDM in approach. He assumed inconsistency as factual marginal probability. in the the DT DT S S between approach. He assumed inconsistency as the the cancellation phenomena do not take place because of their discrepancy the pattern of diagnostic test results in the DT S approach. He assumed inconsistency as the factual marginal probability. factual marginal probability. discrepancy between the pattern of diagnostic test results discrepancy between the pattern of diagnostic test results factual marginal probability. and signatures of the single faults.ofIndiagnostic case of inconsistency, discrepancy between the pattern test results signatures of faults. In inconsistency, The assumption regarding residual cancellation is not the and andassumed signatures of the single faults.regarding In case case of ofmultiple inconsistency, he that the single hypothesis faults signatures of the single faults. In case of inconsistency, The regarding residual is The assumption assumption regarding residual cancellation cancellation is not not the the and he assumed that the hypothesis regarding multiple faults case in consistency based diagnostic methods principally he assumed that the hypothesis regarding multiple faults The assumption regarding residual cancellation is not the is credible. Moreover, he assumed that the signatures he assumed that the hypothesis regarding multiple faults case in consistency based diagnostic methods principally case in consistency based diagnostic methods principally is credible. Moreover, he assumed that the signatures targeting fault isolation focussed on reasoning from the is credible. credible. faults Moreover, he assumed assumed that the signatures case in consistency based diagnostic methods principally is of multiple are composed from the signatures of Moreover, he that the signatures targeting fault focussed reasoning targeting fault isolation isolation focusseddeon onKleer reasoning from the the of faults are composed the signatures of first principles: (Reiter (1987); and from Williams of multiple multiple faults areassumptions composed from from the signatures of targeting fault isolation focussed on reasoning from the single faults. These were also adopted for of multiple faults are composed from the signatures of first (Reiter Kleer and Williams first principles: principles: (Reiter (1987); (1987); deuse Kleer andknowledge Williams the single faults. These assumptions were also adopted for (1987)). These approaches makede of the single faults. These assumptions were also adopted for first principles: (Reiter (1987); de Kleer and Williams development of the multiple fault isolation method single faults. These assumptions were also adopted for (1987)). These make (1987)). These approaches approaches make use of the the knowledge knowledge of method of the structure and behavior of use the of diagnosed system the the development development of the the multiple multiple fault isolation isolation method (1987)). These approaches make use of the knowledge based on binary diagnostic matrix fault presented by Ko´ scielny the development of the multiple fault isolation method of the structure and behavior of the diagnosed system of the structure and behavior of the diagnosed system based on binary diagnostic matrix presented by Ko´ sBarty´ cielny for fault isolation. Generally these approaches e.g. Rebased on binary binary diagnostic matrix presented presented by by Ko´ cielnys of the structure and behavior of the diagnosed system based et al. (2012) and for its modification presented on diagnostic matrix by Ko´ sscielny for isolation. Generally these e.g. Refor fault fault isolation. Generally these approaches approaches e.g. and Re- et al. (2012) and for its modification presented by Barty´ ss iter (1987); de Kleer and Williams (1987); Nyberg et al. (2012) and for its modification presented by Barty´ for fault isolation. Generally these approaches e.g. Re(2014b). et al. (2012) and for its modification presented by Barty´ s iter de Williams Nyberg iter (1987); (1987);(2003) de Kleer Kleer and Williams (1987); Nyberg and the minimal setsand of (2014b). Krysander are and searching for (1987); (2014b). iter (1987); de Kleer and Williams (1987); Nyberg and (2014b). Krysander are for the sets Krysander (2003) are searching searching forobserved the minimal minimal sets of of Nyberg (2006) adopted, to some extent, the idea of dyfaults which(2003) are consistent with the symptoms. Krysander (2003) are searching for the minimal sets of Nyberg (2006) to some the of Nybergdiagnostic (2006) adopted, adopted, to the some extent, the idea idea of dydyfaults which are consistent with the observed symptoms. namic matrix in fastextent, algorithm of multiple faults which are consistent with the observed symptoms. Nyberg (2006) adopted, to some extent, the idea of dyfaultspaper whichconcerns are consistent observedand symptoms. diagnostic matrix in the fast algorithm of multiple This mainlywith the the theoretical practical namic namic diagnostic matrix in the fast algorithm of multiple fault isolation. Instead of binary diagnostic matrix, Nyberg namic diagnostic matrix in the fast algorithm of multiple This concerns mainly This paper paper concerns mainly the theoretical and practical practical aspects of fault location withthe thetheoretical Dynamic and Binary Diag- fault fault isolation. isolation. Instead Instead of of binary binary diagnostic diagnostic matrix, matrix, Nyberg Nyberg This paper concerns mainly the theoretical and practical fault isolation. Instead of binary diagnostic matrix, Nyberg aspects of fault location with the Dynamic Binary Diagaspects of fault location with the Dynamic Binary Diagaspects of fault location with the Dynamic Binary Diag2405-8963 © 2015, IFAC (International Federation of Automatic Control) Copyright 2015 IFAC 1297Hosting by Elsevier Ltd. All rights reserved. Copyright 2015 IFAC 1297 Copyright ©under 2015 responsibility IFAC 1297Control. Peer review© of International Federation of Automatic Copyright © 2015 IFAC 1297 10.1016/j.ifacol.2015.09.704

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applied non-homogenous diagnostic matrix with uncertain fault symptoms and certain lacks of symptoms. However, in Nyberg’s paper he focussed his attention on the development and tests of the fast algorithm of multiple fault isolation according to the parsimony principle rather then on the theory and transformations of dynamic diagnostic matrices. The motivation of this paper is quite practical. The main objective was to deliver the theoretical framework making allowance for ”re-engineering” of BDM d based approaches. The paper contributes in theory development as well as delivers effective multiple fault isolation algorithm intended particularly on-line embedded diagnostic systems. In this paper, we will try to establish, develop and systematize the principles of diagnostic reasoning within the dynamic binary diagnostic matrices. A new view on the inconsistency of the BDM matrix will be given together with appropriate sufficient and necessary conditions of BDM inconsistency. Additionally, the inconsistency of the dynamic binary diagnostic matrix (BDM d ) will be introduced and a theorem of invariancy of the transformation BDM into BDM d matrix in respect to multiple fault isolation will be proven. In this scope, this paper may be considered in the category of continuation and development of the original idea of BDM d . The remaining part of this paper is structured as follows. After brief introduction to the problem of the fault isolation given in Sect. 2, the basic assumptions, definitions and statements regarding dynamic binary matrix BDM d are presented in Sect. 3. Hereinafter, in Sect. 4 the problem of inconsistency of BDM and BDM d matrices is discussed. Sect. 5 introduce some basic operations on BDM d used in the further course of the paper, while Sect. 6 is devoted to the problem of single and multiple fault isolation with BDM d . An algorithm of multiple fault isolation is shown in Sect. 7, and finally the conclusion section completes this paper.

agnostic signals are bi-valued then the matrix V is called as a binary diagnostic matrix BDM (Korbicz et al. (2004)). For convenience, the diagnostic matrix V will be further represented as a block matrix of n vectors Vi : V = [Vi ]n×1 , (3) such that each vector Vi contains the set of diagnostic signals values associated exclusively with the single i-th fault: (4) Vi = [v1,i , v2,i , ... , vm,i ]T . The specific vector Vi that contains all diagnostic signals associated with a particular single fault builds up a numerical or symbolic signature (pattern) of this fault. Consider a model based diagnostic system (F DI) for which a binary diagnostic matrix is given. Let the output of this system in time instant t be a set of residual values Rt = {rjt : j = 1..m}. Now, let the set of residuals Rt be transformed into the set of binary values of diagnostic signals V t = {vjt : j = 1..m} in the process referred to as binary evaluation of residuals. The ordered set of diagnostic values V t captured in the time instant t will be referred to as the vector of actual diagnostic signal values in time instant t or more briefly as a vector of actual diagnostic signal values. The vector V t can match one or even more single fault signatures of the diagnostic matrix. This matching is assumed as indicative to the location of single faults. But, please note that the vector of actual diagnostic signals values V t might be assumed as indicative for multiple faults as well, for example if the vector V t does not match any single fault signature. Hence, in a case that the vector of actual diagnostic signals values V t matches the single faults then the multiple faults are not considered nor searched (Ko´scielny (1995); Ko´scielny et al. (2012); Barty´s (2014a)). The mismatch between a vector of actual diagnostic signals values V t and any single fault signature in a BDM should be commented a little further. The following cases are possible: (1) (2) (3) (4)

2. BASIC ASSUMPTIONS Let us consider a model based diagnostic system. Let a finite set F of n faults fi in this system be defined as: F = {fi : i = 1..n} (1) and a finite set of m diagnostic signals S: S = {sj : j = 1..m}. (2) Let us now discuss the relation between both sets in the form of Cartesian product RF S ⊆ F ×S. Here, the relation RF S constitutes a set of n · m ordered pairs (bi-element relations) fi , sj . According to the geometrical interpretation of the Cartesian product, the relation is the set of n · m points of the plane defined in F and S coordinates. It is possible to spread out a three-dimensional mesh over the F × S plane by attributing diagnostic signal values vj,i of all diagnostic signals sj for all fi faults. The threedimensional mesh is easily transformable into the form of a two-dimensional m · n matrix V of the diagnostic signal values vj,i . The matrix V forms a fault isolation system structure (Gertler (1998)). More comprehensively, this matrix is referred to as a diagnostic matrix. If the di-

there there there there

is no fault; are unknown faults; is single or multiple fault; are multiple faults.

Case (1) is quite trivial. It takes place when all values of the actual vector of diagnostic signals are equal to 0. But allzeroes-vector V t does not exclude a case of unknown faults. Case(2) indicates poor design of the diagnostic system and will not be further discussed. Case (3) may indicate either incidental strong disturbance influencing F DI system or influence of residual cancellation effect. Case (4) is the main concern of this paper. In this paper we will take into consideration the following set of basic assumptions:

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(1) a bi-valued diagnostic matrix of the diagnosed system is known; (2) all single fault are defined; (3) the signatures of all single faults are known; (4) the residual cancellation effects do not take place; (5) the signature of a multiple fault is the logical alternative of the single fault signatures.

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The above given assumptions are acceptable to some extent because in practice the residual cancellation effects are rather exceptional and have marginal meaning. Also, the diagnosed systems having duly prepared diagnostic matrices are not exceptional.

Observation 3. As results from (5), a vertical compression of the binary diagnostic matrix does not influence the count N of all possible combinations of single fault signatures. Therefore, it does not make sense or at least its application should be considered for multiple fault isolation approaches based on signature matching.

3. THE DYNAMIC BINARY DIAGNOSTIC MATRIX

Definition 1. The dynamic binary diagnostic matrix is a matrix resulting from a bi-dimensional structural transformation of any binary diagnostic matrix according to following rules: (1) each i-th column of BDM matrix is removed if (vjt = 0) ∧ (vj,i = 1) and (2) each j-th row of BDM matrix is removed if vjt = 0.

The starting point of the multiple fault isolation based on matching single fault signatures or alternatives of these signatures in BDM with the vector V t of actual values of diagnostic signals, is a quite a trivial constatation that the maximal fault multiplicity does not exceed the number n of single faults. Therefore, searching for multiple faults might be arranged as an iterative process of searching for single, double, triple, ... , and n–multiple faults. In fact, each search needs to identify all combinations of single fault signature which match the current vector of a diagnostic signal values V t . Hence, the total number of all possible combinations of all single fault signatures equals: (5) N = 2n − 1. From (5), it follows that the number of searches for multiple faults grows exponentially with the cardinality of the set of faults F . Therefore, the substantial drawback of this approach is a huge computational effort that must be paid to find out all multiple faults particularly in the case of complex systems where n might be very large. On the other hand, its most important advantage is its extremely easy implementation. Conclusion 1. The efficiency of a multiple fault isolation approach based on matching of signatures depends on the number of columns of the BDM matrix. It is advantageous to note that any zero value of any item of the vector of actual diagnostic signal values V t in the j -th row cannot be alternative of these BDM signatures in which corresponding values in j -th row are equal to 1. Therefore, the binary diagnostic matrix might be compressed dynamically in horizontal direction by rejecting all its Vi columns for which holds: m    vj,i ∧ ¬vjt = 1; ∀i = {1, .., n} . (6) j=1

where: ∧ – is the symbol of Boolean alternative.

This in general, dynamically selects a subset of all signatures of the BDM that must be checked in order to isolate multiple faults. Let the symbol c denotes the number of irrelevant signatures (columns) of the BDM . Hence, the number of combinations of all remaining single fault signatures in relation to the number N from (5) will drop down approximately 2c times : 2n − 1 η = n−c ≈ 2c . (7) 2 −1 Observation 1. The horizontal compression of the binary diagnostic matrix is beneficial in that sense that it allows to speed up the multiple fault isolation. Observation 2. The binary diagnostic matrix may be further vertically compressed . It is easy to see that irrelevant are all the rows of the BDM for which holds: ∀j = {1, .., m} . (8) vjt = 0;

The structural transformation defined above will be denoted as: BDM ⇒ BDM d . Proposition 1. The transformation BDM ⇒ BDM d is invariant in respect to multiple fault isolation. Proof. Each non-zero value of the actual diagnostic signal vjt is assumed as indicative symptom of the single or multiple fault. On the contrary, the zero value of a actual diagnostic signal is assumed as not indicative (unresponsive) to single or multiple fault independently whether a fault or faults really exist or not. Therefore, if the current value of the j -th diagnostic signal vjt = 0 then rejection of the j-th row of the BDM matrix is invariant in respect to the knowledge about faults encoded in non-zero elements of the vector V t . If we consider the assumption (5) from  Sect. 2 than n obviously for any (vjt = 0) the condition ( i=1 vj,i = 0) must hold. Therefore, if this condition does not hold, then it means, that all columns of the BDM for which (vj,i = 1) are irrelevant for dynamic multiple fault isolation. Therefore, all columns of the BDM matrix for which holds (vjt = 0) ∧ (vj,i = 1) may be removed without the loss of information regarding multiple faults. Hence, the transformation BDM ⇒ BDM d is invariant in respect to multiple fault isolation.  The Prop. 1 regarding structural transformation might be generalized if we note that: Proposition 2. The transformation BDM ⇒ BDM d is invariant in respect to multiple fault isolation even if any or both conditions of Def.1 are not met. Proof. If both conditions of Def. 1 are not met then BDM d ≡ BDM and obviously the transformation BDM ⇒ BDM d is invariant in respect to multiple fault isolation. If the first condition is not met than the number of possible multiple faults in BDM d will be the same as in the BDM because the number of columns in both matrices will be identical. The removed rows in BDM d for which (vjt = 0) are uninformative in respect to fault isolation in this sense that the number of multiple faults depends only of the BDM rows for which (vjt = 1).  Conclusion 2. There are maximum 2m transformations of a binary diagnostic matrix into the dynamic binary diagnostic matrices depending on actual vector of diagnostic signal values.

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The structural transformation BDM ⇒ BDM d might be extended by the structural transformation of the vector of diagnostic signals V t ⇒ V d . Definition 2. The dynamic vector of actual binary values of the diagnostic signals V d is a vector resulting from the one-dimensional structural transformation of any binary diagnostic vector of actual values of the diagnostic signals V t by successively removing its zero elements.

From Def. 2 one may conclude that: Conclusion 3. For any non-empty BDM d , the dynamic vector of actual binary values of the diagnostic signals is an all-ones-vector. Example 1. Let us consider a BDM matrix and vector V of the current diagnostic signal values depicted on the left hand side of Tab. 1. The dynamic binary diagnostic matrix BDM d is depicted in the right hand matrix in Tab. 1. This matrix does not contain row s3 because (v3t = 0) and does not contain column f2 because (v3t = 0) ∧ (v3,2 = 1). The dynamic vector of diagnostic signals V d is a vector of ones. Furthermore, in this paper, the characteristic vector V d will be omitted.  Table 1. An example of a structural transformation BDM ⇒ BDM d t

S/F s1 s2 s3 s4 s5

f1 1 0 0 0 0

f2 0 0 1 0 1

f3 1 0 0 1 0

f4 1 1 0 1 1

Vt 1 1 0 1 1

⇒

S/F s1 s2 s4 s5

f1 1 0 0 0

f3 1 0 1 0

f4 1 1 1 1

Vd 1 1 1 1

4. THE INCONSISTENCY 4.1 The inconsistency of BDM The ability of fault isolation directly depends on the structural consistency of the BDM . Definition 3. Any BDM is intrinsically inconsistent if it is empty or any of its columns or rows is an all-zeroes vector. The intrinsically consistent BDM matrix might be inconsistent with respect to the vector of actual diagnostic signal values V t . The sufficient condition of inconsistency of BDM in a narrower sense is as follows: Definition 4. The intrinsically consistent BDM is inconsistent with a vector of diagnostic signal values V t in a narrow sense if it does not contain any single fault signature equal V t . Vi = V t ; ∀i = {1, .., n}. (9) The sufficient condition of BDM inconsistency might be further tightened. Definition 5. The intrinsically consistent BDM is inconsistent with a non-all-zeroes vector of diagnostic signal values V t in a broader sense if there does not exist any Boolean alternative of any single fault signatures which is equal to V t . N (p)  p Ci=1 (Vi ) = V t . (10)  p=1

p (Vi ) is any Boolean alternative of p single fault where: Ci=1 signatures; N (p) = 2p −1 is the number of all combinations of p single fault signatures. 

4.2 The inconsistency of BDM d Definition 7. Any BDM d is inconsistent if and only if it is empty or none of the alternatives of its columns is an all-ones vector. N (p)  p Ci=1 (Vi ) = [1]md ×1 (11) BDM d = {} ∨  p=1

Inconsistent BDM might results from inconsistent BDM d . For example: Proposition 3. Any inconsistent BDM matrix produces an empty BDM d matrix if: n  (12) ∃ j : ( vj,i = 1) ∧ (vjt = 0); j ∈ {1, ..m} d

i=1

Proof. Condition (12) is satisfied if all entries of any BDM row are equally 1 and vjt = 0. In this case all rows of the BDM and all columns will be removed during structural transformation BDM ⇒ BDM d in accordance with assumption (2) of Def. 1. Therefore, the resulting BDM d become empty.  According to conclusion 3, the all-ones-vector V d is a dynamic vector of actual binary values for any non-empty BDM d . Therefore, the necessary and sufficient condition of BDM d inconsistency (11) might be reformulated in a more useful form compared with Def. 7. Definition 8. Any BDM d is inconsistent if and only if it is empty or alternative of its columns is not an all-ones vector. nd  d BDM = {} ∨ Vid = [1]md ×1 (13) i=1

where: n – is a number of columns of the BDM d , Vid – is the i–th column vector (signature) of the BDM d and [1]md ×1 is an md –element of all-ones-vector V d . d

Proposition 4. Any non-empty BDM d matrix is inconsistent if at least one of its row exists for which : d

∃

n 

i=1

d vj,i = 1,

(14)

d where: vj,i is any element of the BDM d .

Proof. From (13) follows the non-empty BDM d is inconsistent if at least one of its rows for which the alternative of all values of BDM d elements equals 0 and obviously is different from 1 exists.  Proposition 5. Non-empty inconsistent BDM d matrix contains at least one all-zeroes-row.  nd d = 0. Proof. Condition (14) is not satisfied if ∃ i=1 vj,i Hence, inconsistent BDM d matrix contains at least one all-zeroes-row.  Proposition 5 is practical because:

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Conclusion 8. In order to check inconsistency of the nonempty BDM d it is enough to check BDM d for any allzeros-row. Example 2. Let us firstly consider a BDM matrix and a vector V t of the current diagnostic signal values depicted in the left hand side of Tab. 1. This matrix is intrinsically consistent because it is not empty and does not contain any all-zeroes column. This matrix is also consistent in a narrower and broader sense because one of its fault signatures (f4 ) matches a vector of actual diagnostic signal values V t . Please note that the dynamic binary diagnostic matrix BDM d depicted in the right hand matrix in Tab. 1 is also consistent because it is not empty and does not contain any all-zeros-row. Example 4. Let us now examine a BDM matrix and a vector V t of the current diagnostic signal values as depicted in the left hand side of the Tab. 3. This matrix is inconsistent in a narrower sense and a broader sense because resulting BDM d matrix contains at least one allzeroes-row. Table 2. An example of inconsistent BDM matrix in a narrower and broader sense. S/F s1 s2 s3 s4 s5

f1 1 0 1 0 0

f2 0 0 0 1 1

f3 1 0 1 1 0

f4 1 0 0 0 1

Vt 1 1 0 0 1

⇒

S/F s1 s2 s5

f2 0 0 1

f4 1 0 1

Vd 1 1 1



Proposition 7 is very useful in practice, particularly if the computational complexity of the applied multiple fault isolation algorithm heavily depends on the number of rows of the BDM d matrix. 6. MULTIPLE FAULT ISOLATION 6.1 The fault-free behavioural dynamic state of the system The fault-free behavioral dynamic state of the diagnosed system is its temporarily diagnostic state in which hypothesis regarding fault-free state is the most credible. The empty BDM matrix is diagnostically useless because it does not give any information regarding relation of fault-symptoms. Therefore, the empty BDM cannot be indicative for the fault-free behavioural dynamic state of the system. For this reason the empty BDM matrices will not be further considered. Proposition 8. The necessary condition for the indication of the fault free dynamic behavioral state of the diagnosed system is the emptiness of BDM d matrix. Proof. Having regard the assumptions depicted in Sec. 2, the fault-free behavioral state of the diagnosed system is indicated by all-zeroes vector of diagnostic signal values V t . Obviously, the non-empty BDM is inconsistent with the all-zeroes vector of diagnostic signals V t . In this case, the transformation BDM ⇒ BDM d produces empty BDM d . Therefore, the empty BDM d matrix might be indicative for the fault free dynamic behavioral state of the diagnosed system.  6.2 The isolation of single faults

5. BASIC OPERATIONS In this section, we will define two useful basic operations regarding expansion and rejection of the BDM d matrix rows. Proposition 6. The expansion of the BDM d matrix by addition of redundant rows is invariant in respect to the number and type of faults. Proof. The hypothetic multiple faults might be assumed as specific aggregates of single faults. These faults are explicitly indicated by the nonzero entries of each BDM d matrix row. Therefore, each j -th row of the BDM d matrix points all hypothetic k –multiple faults for k ∈ {1, ..ndj }, where the ndj is number of ones in this row. Obviously, the number of all combinations without repetitions from the ndj elements is equal to the number of all combinations without repetitions from ndj+1 elements belonging d to the redundant row. Since for redundant rows, vj,i = d d vj+1,i ; ∀i ∈ {1..nj } than any two redundant rows indicate the same number and the same combinations (types) of k -multiple faults.  Proposition 7. The rejection of the redundant rows of the BDM d matrix is invariant in respect to the number and type of multiple faults. Proof. This feature of the BDM is derived directly from the Prop. 6. As the redundant row in the BDM d is not informative in respect to the number and type of multiple faults it might be removed without loss of information regarding the number and kind of multiple faults.  d

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The isolation of single faults with the BDM d is quite trivial. It is sufficient to note, that all dynamic fault signatures Vid of all single faults in BDM d are all-onesvectors. Proposition 9. Any single fault fi is indicated in an nonempty consistent BDM d if and only if its dynamic fault signature Vid is all-ones-vector : Vid = [1]md ×1

(15)

Proof. In accordance with condition (2) of Def. 1, the dynamic vector of actual binary values V d values for nonempty consistent BDM d is an all-ones-vector. Therefore, this vector is indicative for all these single faults for which their dynamic fault signatures Vid are also all-ones-vectors. Clearly, the conjunction of all binary entries of all-onesvector equals 1.  Please note, that if none of columns of the BDM d do not comply with the condition (15) then single faults do not exist. Therefore, according to assumptions adopted in Sect. 2, the hypothesis regarding multiple faults is credible. The possibility of unknown faults or compensation of residuals is excluded. Condition 3. Each i -th column of BDM matrix is removed for which (vjt = 1) ∧ (vj,i = 0) holds. 6.3 The isolation of multiple faults Firstly, we formulate very simple, necessary and sufficient condition of multiple faults in the single row BDM d structure.

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Proposition 10. Any non-empty and consistent single row BDM d is indicative in respect to multiple faults if the number of its non-zero elements (nd1 > 1). Proof. If a specific BDM matrix counts only one row, then at least one of its entry equals 1 because this matrix is considered as consistent and non-empty. Hence, the BDM d matrix indicates at least one single fault. If the number of non-zero entries nd1 > 1 then the number of indicated single faults equals nd1 and the number K m of d multiple faults is equal to Km = (2n1 − nd1 − 1). Hence, K m > 0 for nd1 > 1.  d Proposition 11. Any non-empty consistent BDM is indicative in respect to multiple faults if the number of nonzero elements in any of its rows (ndi > 1). Proof. Any non-empty consistent single row BDM d under condition (ndi > 1) is indicative in respect to multiple faults in accordance with Prop. 10. Therefore, each additional row might increase, or stay unchanged however will not decrease in the number of previously indicated multiple faults.  Proposition 12. The hypothesis regarding multiple faults indicated in the consistent BDM d matrix is credible if:   Vid = [1]1×m∗ ; ∀i = 1, .., nd (16) d

Proof. Let us assume that a single fault will not be considered as a multiple fault. Therefore, if none of the dynamic signatures Vid matches vector V d than a hypothesis regarding single faults is false and a hypothesis regarding multiple faults should be verified. 6.4 Multiple fault isolation algorithm with the BDM d The algorithm of multiple fault isolation with the BDM d summarize the discussion presented in this paper. It is shown in the form of a script of intuitively understandable self-explanatory artificial comment-like metalanguage. Algorithm 1. Multiple fault isolation with a BDM d

Input : BDM : Binary Diagnostic Matrix V : Vector of actual values of diagnostic signals Output: SMF : Set of isolated multiple faults ----------------------------------------------------------(B) BEGIN (*) Clear the set SMF of dynamic isolated faults. (*) Is V an all-zeroes-vector? (*) Finish algorithm with the status {error-free} if V is an all-zeroes-vector. Go to the END. (*) Transform BDM into dynamic BDM_d matrix. Get: m_d,n_d. (*) Check consistency of the BDM_d matrix. (*) Is the BDM matrix consistent? (*) Finish algorithm with status {erroneous BDM} if BDM_d is inconsistent. Go to the END. (*) Transpose BDM matrix into BDM_d matrix. Get: m_d,n_d (*) Is the number of BDM_d rows less then the number of columns? (*) If true, fill in the BDM_d adding successively the copy of any row to this matrix until m_d=n_d. Go to (D). (*) Reject redundant rows in BDM_d leaving at least n_d rows. (D) Decompose BDM_d matrix on multiple faults rejecting redundant multiple faults. Get set of SMF. (E) END

7. CONCLUSION This paper to some extent closes the gap in the theory of dynamic binary diagnostic matrices. In practice, it provides ”re-engineered” and theoretically well established multiple fault isolation algorithm. It was shown that dynamic bi-directional compression of BDM is invariant in respect to fault location and fault type. Reducing the size of the BDM matrix has hope for a much better computational effectiveness of the fault isolation algorithms applied so far. This has a crucial meaning in the on-line applications of the diagnostic systems. REFERENCES Barty´s, M. (2014a). Chosen Issues of Fault Isolation. Polish Scientific Publishers PWN. Barty´s, M. (2014b). Multiple Fault Isolation Algorithm Based on Binary Diagnostic Matrix, volume 230 of Intelligent Systems in Technical and Medical Diagnosis, 441–452. Springer, Heilderberg. Barty´s, M. (2014a). Comparative study of the three chosen multiple fault isolation algorithms. Journal of Physics: Conference Series. (in printing). Blanke, M., Kinnaert, M., Lunze, J., and Staroswiecki, M. (2006). Diagnosis and Fault Tolerant Control. Springer Verlag, New York. de Kleer, J. and Williams, B. (1987). Diagnosing multiple faults. Artificial Intelligence, 32(1), 97–130. Gertler, J. (1998). Fault Detection and Diagnosis in Engineering Systems. Marcel Dekker Inc. Isermann, R. (2006). Fault Diagnosis Systems: An Introduction from Fault Detection to fault Tolerance. Springer Verlag, Heidelberg. Korbicz, J., Ko´scielny, J.M., Kowalczuk, Z., and Cholewa, W. (2004). Fault Diagnosis. Models, Artificial Intelligence, Applications. Springer, Berlin. Ko´scielny, J.M. (1995). Fault isolation in industrial processes by dynamic table of states method. Automatica, 31(5), 747–753. Ko´scielny, J.M., Barty´s, M., and Syfert, M. (2012). Methods of multiple fault isolation in large scale systems. IEEE Transactions On Control Systems Technology, 20(5), 1302–1310. Nyberg, M. (2006). A fault isolation algorithm for the case of multiple faults and multiple fault types. 679–689. 6-th IFAC Safeprocess Symposium, Beijing, PR China. Nyberg, M. (2011). A generalised minimal hitting-set algorithm to handle diagnosis with behavioral modes. IEEE Transactions on Systems, Man, and CygerneticsPart A: Systems and Humans, 41(1), 137–148. Nyberg, M. and Krysander, M. (2003). Combining ai fdi and statistical hypothesis–testing in a framework for diagnosis. 891–896. 5th IFAC Symposium Safeprocess, Washington DC, USA. Patton, R., Frank, P., and Clark, R. (2000). Issues of Fault Diagnosis for Dynamic Systems. Springer Verlag, New York. Reiter, R.A. (1987). Theory of diagnosis from first principles. Artificial Intelligence, 32(1), 57–95. Trav´e-Massuy´es, L. (2014). Bridges between diagnosis theories from control and AI perspectives, volume 230 of Intelligent Systems in Technical and Medical Diagnostics, 441–452. Springer, Heidelberg.

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