AP07_1.vp 1 Introduction The approaches to unexpected situations (as a special class of so-called undetectable faults) come from various domains, and they are referenced, e.g., in [1], [2], [3], [4], [5], [6], [7], [8], [9]. (A detailed analysis of these sources is in [17].) Our approach for UX3 detection is based on the concept of a UX3 type situation, on the concepts of a Model of a Sys- tem of Situations (MSS) and a Model of a System of Faults (MSF), and on an original method which detects a UX3 type unex- pected situation as a violation of a proper structural invariant - constructed on MSS (MSF). The structural invariant is con- structed on MSS during the so-called “cognitive phase” of MSS (MSF) development. In this “cognitive phase”, it is con- sidered that some “classes” of situations (faults) have already been established but some of the new situations are processed with large uncertainty. Violation of the structural invariant represents the detection of a UX3 type situation. (A more de- tailed explanation is given in section 2). Analysing our approach from the Fault Diagnosis (FDI) point of view, some important issues may be indicated. The first issue is the concept of MSS (MSF) in the context of the Model-Based approach in FDI, and the assignment of our method to some of the known diagnostic approaches, e.g. to abductive diagnostics. Our approach is model-based, though the development and the use of models (MSS, MSF) are different from the examples introduced, e.g., in [10]. Our models are developed as a result of data and signal analysis (not as a re- sult of preformed knowledge about a diagnosed system, e.g., knowledge about the internal structure or about a mathemati- cal model). As will be explained in the following sections, the phase of fault detection in an abductive diagnostic model (e.g., in [11], [12]) is a rather special case of UX3 detection in our method. The second issue is the concept of a symptom. Our approach concentrates on processing situation signals (data) of the fol- lowing types: vectors of outputs from qualitative models, ul- trasound signals (representing the internal structure of the material samples), sequences of symbols (signs) and words from monitoring processes, ECG and EEG signals. Special situations (faults) which represent an extraordinary (faulty) behavior of a monitored (diagnosed) system are spread along the run of the signals (e.g., in their morphology) and in the sets of data. It is sometimes hard to speak about symptoms (symptoms of What?). The presented approach to UX3 detection has been tested by the following three application cases: � In a supervisory control system for an industrial distillation column, especially in a qualitative model designed for the starting phase of the distillation process (e.g., after mainte- nance operations), details, in [13], [14]. � In detecting unexpected faults in welds (laser, micro-arc and electron beam welds of thin walled welded structures used in the aerospace industry) in combination with neural fault detec- tors, [15], [16] � Within the framework of a special supervisory and moni- toring system, e.g., in [17]. 2 Unexpected situations – concepts and examples General features of unexpected situations and three basic classes of unexpected situations will be introduced in this section. The first class (UX1) is induced by the relativity of the un- expected situation in respect to the levels of available Data and Knowledge of a reasoning human. (We will denote these situations as UX (UX1) – emphasizing the intuitive aspect of the detection of such situations.) Example 2.1: Let us suppose that the extent of values measur- able in an instrument (given in the instrument protocol) is umin, umax. The situation when we measure by this instrument a value 10 umax (without problems) is a UX 1 situation. (One interpretation of such a phenomenon is that wrong informa- tion was introduced in the protocol of the instrument.) Example 2.2: The correct representation of a process by a dif- ferential equation depends on the identified type of equation and on the precision of the identified quotients. All cases with unknown noise in the input or in the output variables, un- known drifts in the parameters and cases of so-called hidden parameters, are cases that generate UX1 situations. © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 3 Acta Polytechnica Vol. 47 No. 1/2007 Fuzzy Concepts in the Detection of Unexpected Situations J.Bíla, J. Jura This paper establishes three essential classes of unexpected situations (UX1, UX2, UX3), and concentrates attention on UX3 detection. The concepts of a special Model of System of Situations (MSS) and a Model of a System of Faults (MSF) are introduced. An original method is proposed for detecting unexpected situations indicating a violation of a proper invariant of MSS (MSF). The presented approach offers a promising application for starting and ending phases of complex processes, for knowledge discoveries on data and knowledge bases developed with incomplete experience, and for modeling communication processes with unknown (disguised) communication subjects. The paper also presents a way to utilize ill-separable situations for UX3 detection. The paper deals with the conceptual background for detecting UX3 situations, recapitulates recent results in this field and opens the ways for further research. Keywords: unexpected situations, model of system of situations, model of system of faults, invariants, fuzzy variables, degree of unexpectedness, emergence zone, association rules, Hasse Diagram. This text was a part of the Intenational conference on Advanced Engineering Design (AED 2006), which was held in Prague in 2006. Situations of the second class (UX2) are generated by models that are a priori insufficient for representing some situa- tions in the modeled process or system. (This means, e.g., that most situations are well represented, but a small number of situations are represented incorrectly.) Example 2.3: Situations generated as an unprovable formula in FOL (First Order Language), situations for which a Turing machine does not stop (or works too long), or situations in complex robotic production lines (which are impossible to simulate wholly), belong to this class. Example 2.4: Situations generated by models of systems with deterministic chaos. For example, systems modeled by Duff- ing or Lorenz equations belong to this class. Situations of the third class (UX3) have causes that differ from those that induce situations of the two previous classes. One principal scenario for UX3 emergence is shown in Fig. 1. A general type of model is considered as a pair (C, I) which represents the synthesis of a Carrier (C) and Information (I). (A simple example of such a model is a classical photo- graph, where the photographic paper represents the carrier. What a human interpreter sees in the photograph is the information.) Most cases of sign modeling work with so-called “hard” carriers equipped with resistance to the influence of the infor- mation that is encoded on them. However, sign and symbolic models with hard carriers are not resistant to the encoding and transfer of false information, (though they are very suc- cessful even in commercial terms). Such a case is described in Example 2.5. The whole scenario of this example is im- portant, including the physical and technical background. It is this background that distinguishes this case from a case of ciphering (which lies beyond the scope of our paper). For this reason we search for additional models M3(C3, I3), … , Mn(Cn, In), which enable the correctness of the model M2(C2, I2) to be checked with regard to M1(C1, I1). Such models in most cases really exist, being induced during the evolution of M1(C1, I1) and M2(C2, I2). However, it is not triv- ial to discover such models. The correctness of function M2 with regard to M1 is verified (in cases when some of models M3, …, Mn are discovered) by symbolic commutation of the transformation diagrams (from Fig. 1): h g of h g of1 1 1 1� �, ,� � n n (1) This brief explanation of essential concepts for the theory of UX3 situations introduced above will be extended and supplemented by a formal description and a recent applica- tion of this theory in section 3. Example 2.5: Approximation of UX3 type situations by a scheme with standard intentions. Let us consider the scheme in Fig. 2. The scheme consists of an unavailable (“invisible”) part and a transparent part. The invisible part contains: a process (for which we have no model), process observer variables (x1, …, xn), non-process external variables (y1, …, ym) and switches. The transparent part con- tains: an observer and a situation recognition block with classes of situations (S1, …, Sq) and with the developed MSS. The situation recognition block (which contains MSS) is de- veloped during a standard operation of the process as a result of the work of the observer. The process is “represented” (for the observer) by the process observer variables (x1, …, xn). Let us consider that after a period of successful function, the structure of the situation recognition block is accepted as stable. Now let us assume that some of the switches are suddenly switched to variables (y1, …, ym) which represent another external reality, but they are formally the same as (x1, …, xn). As a result of this action – the assignment of the new situations coming into classes (S1, …, Sq) will work incorrectly. Such a change is undetectable using standard methods, and it requires a special detection approach. Example 2.6: Special UX3 situations emerge in cases when inappropriate intentions (in Frege’s sense, e.g., in [18]) are used to describe a process. (Usual intentions are propositions, quantities or properties.) Quantities, for example, are some- times wrongly used for complex processes (or systems with complicated behavior) which are poorly measurable, and rep- resenting them by time series or by complicated signals intro- duces further difficulties in processing and interpretation. Typical examples of such cases are ECG, HRV (Heart Rate Variability) signals. (These facts are known, and they have 4 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 47 No. 1/2007 M 1 (C 1 , I 1 ) I 2 M 3 (C 3 , I 3 ) M n (C n , I n ) M 2 (C 2 , I 2 ) g 1 … I 3 I n h 1 g n h n f 1 Fig. 1: Models, carriers and information S 1 , …, Sq Situation Recognition Block Observer Process x1 x2 xm ym y2 y1 s1 sj sp Invisible part Transparent part Fig. 2: Approximation of a UX3 type situation been published (e.g., in the Journal of Cardio-Vascular Re- search from 1996), and have been presented in our research (e.g., in [28].)) 3 Formal models for detecting UX3 The method proposed in this paper for detecting UX3 uses a Model of a System of Situations (MSS) or a Model of a System of Faults (MSF). Both these systems are developed in the cognitive phase (as was mentioned in the Introduction) dur- ing operations and experiments with the observed process or with the FDI system. The goal of the “cognitive phase” is to form structural invariants. (A few types of such invariants will be introduced in subsections 3.2–3.4.) In our method, a violation of a structural invariant is a means for detecting a UX3 type situation. (An investigation of the necessary statistical pa- rameters of the “cognitive phase” (e.g., “How many situations need to be analysed and in which classes can they be searched for”, etc.) was made in [26].) In this paper we assume that the “cognitive phase” satisfies the necessary statistical and model- ing standards.) Model MSS has in general the following form: MSS S S S Inv Inv� , 1� � � �( ), , ( ) , ( ), , ( )� �n i p , (2) where S represents a basic set of situations, � �1( ), , ( )S S� n are structures on S considered as relevant for UX3 de- tection and Inv Inv( ), , ( )� �i p� are invariants on some � �1( ), , ( )S S� n � �( , , , )i p n� 1 � for UX3 detection. Model MSF has in general the following form: MSF S F S F S F Inv Inv � , , ,1, ( ), , ( ) , ( ), , ( ) , � � � � � � n i p , (3) where S F, represent basic sets of situations and faults, � �1 , ,( ), , ( )S F S F� n are structures on (S, F) considered as relevant for UX3 detection and Inv Inv( ), , ( )� �i p� are in- variants on some � �1( ), , ( )S S� n for UX 3 detection. Models for UX3 detection have the forms MD MSS COND MD MSF COND Inv Inv ( ) ( ) , UX , UX , V V 3 3 � � or (4) where CONDVInv represents the conditions of violation of MSS (MSF) invariants. (These conditions are analysed in the process of UX3 detection.) Fig. 3 illustrates the position of the Models for UX3 de- tection in a block scheme of the FDI system respecting one of many possible structures of the FDI system. The figure ex- presses only the fact that MSF and MD(UX3) work as a paral- lel block with a Fault Recognition System (which is usually un- derstood as an ending member of an FDI system). 3.1 A general type of structural invariants Inv(�i ), …, Inv(�p ) The general type of invariants Inv(�i), …, Inv(�p) is con- nected with the commutation of diagrams (1) and is limited in this paper to the form of morphisms hi, gi, for i � 1, …, n: M C I g M C I M C I h i i i i i i i� � � � � �� �2 2 2 2 2 2 2 2 2( , ) ( ( , )), , ( , )�� i M C I( ( , )).1 1 1 (5) The concrete form of the morphisms depends on the type of models M C I1 1 1( , ) and M C I2 2 2( , ). The following subsections introduce three examples of MSS and MSF and MD(UX3). 3.2 MD(UX3) with an emergence zone MSS has (in this case) the form (6) MSS S M S� � � � � � � , ( ), , , ( , ( ( , ) ))s s� � , (6) where S represents a basic set of situations, M(S) is a matroid constructed on this set of situations, � is a Basis on M(S), � represents a Cover on M(S) (a subset of the matroid closure), �( , )s � is a metric function and � is a positive real number. In addition to � �, there is the so-called emergence zone �. In this zone there are elements that can not be constructed from the elements in � �, , but these elements are relevant to MSS and they are not in zone Outside, see Fig. 4. With regard to the fact that “regular” situations can be assigned in � or in � or they are classified in zone Outside (law of the excluded third alternative), elements of � are considered as extraordinary situations and they represent (in the context of our paper) UX3 situations. In this case MD(UX3) has the form (7) MD MSS MD MSS ( ) , ) ( , )) ( ) ) UX U UX p 3 3 ,( (s or , ( s � � � � � � � � � � (7) where s* is an unexpected situation and Up is a real number (Upper bound). © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 5 Acta Polytechnica Vol. 47 No. 1/2007 Process (System) Sensors Diagnostic Model Extraction of Diagnostic Situations Fault Recognition System Detection of Unexpected Faults Fig. 3: Position of Models for UX3 detection in a rough block scheme of the FDI system M S( ) o s* Outside � Fig. 4: Basis, cover and emergence zone The MSS described above was used in real conditions for UX3 detection in the starting phase of a deisobutanisation process [13], [14]. 3.3 MD(UX3) with bipartite graphs In this case, MSF has the form (8) � �MSF S F G G G Gm� , , , , »1 2, , ,� , (8) where (S, F represent basic sets of situations and faults, G is a bipartite graph (Situation � Faults), � �G G Gm1 2, , ,� , » is a special Dulmage-Mendelsohn decomposition, [19], [21]. ( , , ,G G Gm1 2 � are irreducible sub-graphs, “»” is a tree order- ing on the set of the sub-graphs.) The detection of UX3 has been represented by a violation of ordering “»” and has been indicated by the following conditions discovered for a situa- tion s*. In this case, MD(UX3) has the form (9) MD MSF ACC ACC( ) ,((( ( , )) ( ( , ))),UX G s AND not G s for 3 � � �i j some G » Gj j i( )) , (9 ) where s* is an UX3 related to � �G G Gm1 2, , ,� , » . Note 3.1: Expression ACC(G, s) denotes “situation s is ACCepted by bipartite graph G”. This MD(UX3) model has been used, e.g., in [15] and [16]. (However, taking into account the well-studied concept of DM-irreducibility (e.g., in [20], page 63, 64) we are aware of the limited applicability of DMD.) 3.4 MD(UX3) with the association rules The background for MD(UX3) model presented in this section continues in the line started in [22],[23] and nowadays utilises formulations, e.g., from [24], [25]. In this case, MSF has the form (10) MSF S F M HD ER� , , , ,G , (10) where (S, F represent basic sets of situations and faults, MG is a qualitative matrix with data acquired from the cognitive phase of FDI system operation (the concept “cognitive phase” was introduced and explained in section 1.), HD is a Hasse Diagram [25] derived from MG and ER is a set of Evaluated Association Rules extracted from HD (each rule evaluated by quantities of Supp (Rule Support) and Conf (Rule Confi- dence), [24]. Note 3.2: The Hasse Diagram facilitates the process of extracting the rules from matrix MG, but its use is not obligatory for the formation of ER. MD(UX3) has (in this case) the form (11) MD MSF ER ACC( ) , ( , ( , ))UX not s3 � � � �r r (11) where s* is a UX3 situation. Note 3.3: The expression ACC(r, s*) denotes “situation s* is ACCepted by rule r”. 3.5 Additional important circumstances A. The basic purpose of MSS (MSF) and of MD(UX3) is in the formation of rules that enable to distinct between “ill-separa- ble situations” and “UX3 situations”. Such decision rules are simple: �IF situation s satisfies the invariant� � �THEN s it is an ill separable situation . �IF situation s does not satisfy the invariant � � �THEN s is UX3 situations�. B. When detecting a UX3 situation, we should like to know how important the discovered UX3 situation is in comparison with other possible UX3 situations (which could be detected by the considered invariant). For this reason a numerical function D Inv Inv( ( ), , ( ) , )� �i p UX� 3 has been suggested. It is called the Degree of UX3 (the Degree of Unexpectedness) with respect to invariants Inv Inv( ), , ( )� �i p� and depends on the following two factors: � the complexity (the constructibility) of the applied invariant (invariants), � the sensitivity of the invariant (invariants) to the measure of the violation. 4 Degree of UX3 The following form has been introduced for the computa- tion of D(Inv(�p), UX 3) (with one invariant Inv(�p)) D Inv Inv ( ( ), )) ( ( )) ( ( )) � � p i i x UX x i 3 1 2 1 1 � � � �Q cpm p � � ELM� � � � � � � � � � � � � � � 1 1 2 2 a i i i x x x i ( ( ( ) ( ))) ( ( )) � � � ELM Inv � p � � � � � � � � 1 a , (12) where QCpmx is a quotient of the invariant complexity, xi are elements of the invariant structure (from the ground set Elm(Inv(�p)), �i are quotients of importance for elements xi. Quantities 1�(xi) are quotients of deployment of elements xi before invariant violation and 2�(xi) are quotients of deploy- 6 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 47 No. 1/2007 No. Name of Structure QCpmx [1] 1 Klein group of the 4th order 1 2 Permutation group of the 6th order (4×4) 0.9455 3 Group of linear transformations of the �th order 0.7432 4 Semi-group of binary equivalencies 0.5571 5 Semi-group of binary relations 0.1979 6 Matroid of the 2nd order 0.1750 7 Linear regular grammar 0.1345 8 Non context grammar 0.1047 9 Structure of association rules 0.0883 Table 1 ment for elements xi after violation of the invariant by means of xi. (Number a � 2 (usually) but not necessarily). Note 4.1: If an element is not violated, it holds: 1�(xi) � 2 �(xi). Quotient of structure complexity QCpmx expresses the dif- ficulty to form a model of such a structure. The complexity of the structures is compared in [26]. Some illustrative exam- ples of QCpmx quantities for structures with one composition operation are introduced in Table 1. Note 4.2: The specification of structures 1–5, 7, 8 from Table 1 is known from the literature. Structure 6 is a matroid with 10 independent sets and with the cardinality of its basis equal to 2. The quantity of D(Inv (�p), UX 3) gives a qualitative evalua- tion of the difficulty of the operations that follow after UX3 detection. Usual such operations (e.g., in the FDI field) are: localisation, isolation, identification and interpretation a UX3 situation. The higher the D(Inv (�p), UX 3) quantity is, the more difficult these operations are to execute. Example 4.1: Let us suppose Klein group B/ � �{I, N, R, C}, o as an invariant Inv(�1), and let us suppose the violation of ele- ments N and R given by the following quantities of quotients: �I � �N � �R � �C � 1, 1�(xI) � 1�(xR) � 1�(xN) � 1�(xC) � 1, 2�(xI) � 2�(xC) � 1, 2�(xN) � 2�(xC) � 0.8 and a � 2. D( B/ , UX3) � (0.08/4)1/2 = 0.1414. Example 4.2: Let us suppose a set of rules ER (with 11 rules) as an invariant Inv(�1), and let us suppose violation of rules r3 and r5 given by the following conditions: �1 0 7� . , �2 0 7� . , �3 1 4� . , �4 0 9� . , �5 1 0� . , �6 1 2� . , �7 0 5� . , �8 0 8� . , �9 0 5� . , �10 1 2� . , �11 0 7� . , 1 1 1 2 1 11 1� � �( ) ( ) ( )r r r� � � �� , 2 1 2 2 1� �( ) ( )r r� � , 2 3 2 5 0 0� �( ) ( ) .r r� � , 2 4 2 2 7 2 8 2 9 2 10 2 11� � � � � � �( ) ( ) ( ) ( ) ( ) ( ) ( )r r r r r r r6� � � � � � � 1 and a � 2. D ER( , ) , , UX Q Cpmx i i 3 3 2 5 2 2 1 11 1 2 1 � � � � � � � � � � � � � � � � � � � � � 0 565 0 0883 6 4 . . .� . The examples introduced here illustrate the D(Inv(�p), UX 3) quantities for violation of two very different invariant struc- tures. The results correspond to an intuitive understanding of variable D(Inv(�p), UX 3). The more complicated the structure of Inv(�p) is and the higher its violation is, the higher is the potential quantity of D(Inv(�p), UX 3). (For illustration: D( B/ , UX3) � [0, 1], D(ER, UX3) � [0, 11.325].) 5 Conclusions This paper demonstrates the use of a fuzzy approach for modeling very complex problems. Fuzzy concepts are con- tained in all essential conceptual constructs as UX3, MSS, MSF, a violation of an invariant, emergence zone, Evaluated As- sociation Rules, Hasse Diagram (and in the fuzzy values and variables). The paper has introduced methods for UX3 detection. It has introduced a general approach for the development of MSS, MSF and MD(UX3) models, and three variants of these models have been described. Examples of MD(UX3) with an emergence zone, with Bi- partite Graphs and with Evaluated Associations rules in the role of structural invariants of MSS and (MSF) have been pre- sented, e.g., in [13]–[17].) In [17] we introduced an illustrative application of MD(UX3) with Evaluated Association Rules for the conditions of an industrial monitoring system (developed with data sup- port from “Ventilation system of the Mrazovka road tunnel in Prague” in Czech Republic). The proposed method may be applied for processes with a similar formal description, e.g., in transport systems, power supply systems, and in the chemi- cal industry. The approach is also well applicable in special signal based cases (with no available model of the observed and de- tected system), where the signals are acquired from special sensors and especially when neural networks or fuzzy systems are used for processing them. (Such applications fields are described, e.g., in [16], [27], [28].) 6 Acknowledgments This research has been supported by Research Grant MSM No. 2B06023 References [1] Abramovici, M., Breuer, M. A., Friedman, A. D.: Digital System Testing and Testable Design, IEEE Press, New York, 1995, p. 99–104. 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