Graph-Theoretic Analysis of Residual Generation Under Computational Constraints

1 Graph-Theoretic Analysis of Residual Generation Under Computational Constraints Jan ˚ Aslund Abstract —A unified structural framework is pr esented for model-based fault diagnosis that explicitly incorporates both fault locations and constraints imposed by the residual generation methodology . Building on the concepts of proper and minimal structurally overdetermined (PSO/MSO) sets and T est Equation Supports (TES/MTES), the framework introduces testable PSO sets, Residual Generation (RG) sets, irreducible fault signatures (IFS), and Irr educible RG (IRG) sets to characterize which submodels are suitable f or residual generation under gi ven computational restrictions. An operator M ∗ is defined to extract, from any model, the largest testable PSO subset consistent with a specified residual generation method. Using this operator , an algorithm is developed to compute all RG sets, and it is shown that irreducible fault signature sets form the join-irreducible elements of a join-semilattice of sets and fully capture the multiple-fault isolability properties in the method-constrained setting. The approach is exemplified on a semi-explicit linear D AE model, where low structural differ ential index can be used to define M ∗ . The results demonstrate that the proposed framework generalizes MTES-based analysis to residual generation scenarios with explicit computational limitations. Index T erms —Fault detection and isolation, Model-based di- agnosis, Structural methods I . I N T RO D U C T I O N A well-established approach in model-based diagnosis is to exploit redundancy within a system model to design an- alytical tests capable of detecting inconsistencies. Numerous methods have been proposed to identify structurally redundant submodels suitable for this purpose. In [14], [9], algorithms were dev eloped for identifying Minimal Structurally Overde- termined (MSO) sets. The method of Possible Conflicts was introduced in [17], while [20] employed Structural Analytical Redundancy Relations. A comprehensi ve ov erview of various definitions and algorithms can be found in [15]. In addition to redundancy , an important aspect is the methodology employed for test construction. Several studies hav e addressed this issue by proposing structural characteri- zations of equation subsets that are applicable under specific constraints imposed by the residual generation process. In [19], a method for identifying MSO sets for sequential residual gen- eration was proposed, explicitly considering the computational tools available for solving algebraic and differential equations as well as for numerical dif ferentiation. The notion of causally computable MSO sets was later introduced in [18], where computational sequences were de- riv ed under the assumption that linear square subsystems are in vertible, subject to causality constraints among variables. Department of Electrical Engineering, Link ¨ oping University , Link ¨ oping, Sweden (e-mail: jan.aslund@liu.se). In [6], the concept of the monitorable part was defined under deriv ative, integral, and mixed causality assumptions, and the implications for fault isolability were systematically analyzed. Furthermore, [16] presented a method for identifying diagnosable subsystems characterized by their suitability for observer -based residual generation and their ability to achieve maximal fault isolability . The degree of redundancy also plays a critical role in the design of diagnostic tests. For instance, in observer - based approaches, high redundancy is desirable as it enhances robustness to modeling errors and measurement noise. How- ev er, increased redundancy may reduce fault isolability , since residuals derived from larger submodels tend to be sensitiv e to a broader set of faults. A structured approach to address this trade-off was introduced in [13], where the concept of T est Equation Supports (TESs) was defined to account for both the lev el of redundancy and the distribution of faults in the system model. The aim of this work is to integrate the aforementioned perspectiv es through a unified framework that considers both fault locations and the residual generation methodology . The analysis further inv estigates the implications of this integration and establishes an appropriate graph-theoretical basis. I I . T H E O R E T I C A L BA C K G RO U N D Graph theory provides a useful framework for analyzing structural properties of systems and for identifying submod- els with redundancy . Consider a set of equations M = { e 1 , . . . , e m } defined over a set of unknown variables X = { x 1 , . . . , x n } . A bipartite graph can then be constructed with node sets M and X , and an edge set E ⊂ M × X , where ( e i , x j ) ∈ E if the variable x j ∈ X appears in equation e i ∈ M . As an illustrative example, consider the system of ordinary differential equations (ODEs) ˙ x 1 = x 1 + x 3 , ˙ x 2 = x 2 − x 3 , ˙ x 3 = x 1 . The corresponding bipartite graph can be represented by the bi-adjacency matrix   X X X X X X   where an X in position ( i, j ) indicates that ( e i , x j ) ∈ E . An alternativ e representation that explicitly incorporates dif feren- tial constraints is described in [3]. 2 The Dulmage–Mendelsohn (DM) decomposition of bipartite graphs is a powerful tool for structural analysis [3], [4]. The decomposition, illustrated for a general system in Figure 1, permutes the rows and columns of the bi-adjacency matrix so that three distinct parts of the model can be identified: an ov erdetermined part M + with more equations than unknowns ( | M + | > | X + | ), an exactly determined part M 0 , which intro- duces as many new variables X 0 as equations ( | M 0 | = | X 0 | ), and an underdetermined part M − , which introduces more ne w variables X − than equations ( | M − | < | X − | ). M − M 0 M + X − X 0 X + Fig. 1. Illustration of the bi-adjacency matrix in a Dulmage–Mendelsohn decomposition, where all edges lie within the shaded area The ov erdetermined part M + is of particular interest for diagnosis, since it contains redundant information that can be used to detect inconsistencies. A characterization that a matrix is structurally in vertible means that almost all numerical matrices with the same zero–nonzero pattern are inv ertible. Similar structural char- acterizations will be used later in the paper . The following definitions from [14] provide two structural characterizations of redundant subsets that are suitable for the design of diag- nostic tests. Definition 1 (PSO set, MSO set, and redundancy φ ) . A set of equations M is Pr oper Structurally Over determined (PSO) if M + = M . A PSO set is Minimal Structurally Overdetermined (MSO) if no pr oper subset of M is PSO. The r edundancy φ ( M ) of a PSO set M is defined as the differ ence between the number of equations and the number of unknown variables in M . In [13], the location of faults was incorporated into the structural analysis through the following definitions. Definition 2 (TES and MTES) . Let F ( M ) denote the set of faults included in an equation set M . A PSO set M is a T est Equation Support (TES) if F ( M )  = ∅ and for any PSO set M ′ ⊋ M it holds that F ( M ′ ) ⊋ F ( M ) . A TES is minimal (MTES) if no pr oper subset of M is a TES. In the definition of MTESs, fault locations in the model are explicitly taken into account. The main objectiv e of this paper is to generalize these structural concepts by also incorporating the residual generation method into the analysis, and thereby to identify a graph-theoretical representation that captures both fault distribution and computational constraints. For further examples, discussions, and information about implementation of MSO sets and MTESs algorithms, see [10] and [8]. I I I . B A S I C D E FI N I T I O N S The follo wing system of linear equations will be used in the forthcoming sections to illustrate some basic ideas and to highlight the consequences of imposing restrictions on the method used in the design of diagnostic tests. e 1 : x 1 + 2 x 2 = u 1 + v 1 e 2 : x 1 + x 2 = u 2 + v 2 e 3 : y 1 = x 1 + 3 x 2 + v 3 (1) e 4 : y 2 = x 1 − x 2 + f 1 + v 4 e 5 : y 3 = x 2 + f 2 + v 5 Here u i and y i are kno wn signals, x i are unkno wn variables, and the vector v = ( v 1 , . . . , v 5 ) T is gaussian random vector with zero mean and identity cov ariance matrix. The fault signals f 1 , and f 2 , are modeled as additive faults, and note that it is not assumed that all equations include a fault. This is often the case for basic physical equations, or faults that can be discovered by other means and therefore can be handled separately . One example of the latter is sensors with a defined behavior under fault conditions, such as a sensor that is considered healthy when its output remains within a specified fail-safe range and indicates a fault when the signal leav es this range; see e.g. Chapter 12 and 13 in [5]. A. Restrictions on the r esidual generation methodology From no w on, it is assumed that variables in system (1) can only be computed by sequential back-substitution, and that no row operations to eliminate variables are allowed. The purpose of analyzing system (1) under this artificial computational constraint is to mimic situations that may arise, for example, when considering system models with non- in vertible subsystems. A more natural computational constraint will be presented in Section V. B. Residual generation In this section, the possibility to create residual sensitiv e to faults in model (1) is in vestigated under the restriction introduced above. First, the noise signals are omitted and it is assumed that the system operates in the fault-free mode. Consider the v ariable x 2 . Under the computational restriction, the only way to compute x 2 is to use equation e 5 in model (1), x 2 = y 3 . Then x 1 can be computed using, for example, equation e 2 , x 1 = u 2 − y 3 . By substituting x 1 and x 2 into the redundant equation e 1 , the residual r 1 = u 1 − u 2 − y 3 = 0 3 is obtained, which is equal to zero in the fault-free case. Including faults and noise in the model yields r 1 = u 1 − u 2 − y 3 = f 2 + w 1 , where w 1 = − v 1 + v 2 − v 5 . The first expression shows how the residual is computed from the known signals u 1 , u 2 , and y 3 , while the second expression sho ws how the residual depends on the unknown fault f 2 and the noise w 1 , which can be used to e valuate the fault-to-noise ratio of the residual. In this case, the MSO set { e 1 , e 2 , e 5 } was used to construct the residual r 1 . This w as possible since, first, the unknown variables in the submodel could be computed under the imposed restriction and, second, there was redundancy in the model, i.e., the system was overdetermined. These two prop- erties can be read directly from a graph representation of the system and admit a straightforward structural characterization. Definition 3 (Structurally testable) . Let a computational con- straints be specified by a structural characterization of which equation sets can be used to generate r esiduals that ar e sensitive to all faults included in the equation set. A PSO set that satisfies this structural characterization is called structurally testable. Next, it is shown how the fault-to-noise ratio can be improv ed by considering a lar ger PSO set with more redundant equations. As a first step, the MSO set { e 1 , e 3 , e 5 } , instead of { e 1 , e 2 , e 5 } , is used to deriv e the residual r 2 = y 1 + y 3 + u 1 = f 2 + w 2 , where w 2 = v 1 + v 5 − v 3 . The residual has been scaled so that the dependence on the fault f 2 is the same for both residuals. The next step is to form the af fine combination r = k r 1 + (1 − k ) r 2 = f 2 + v , where v = k w 1 + (1 − k ) w 2 , and choose k to minimize the variance of v . It is straightforward to show that the covariance matrix of the random vector w =  w 1 w 2  T is Σ w =  σ 2 11 σ 12 σ 21 σ 2 22  =  3 − 1 − 1 3  . By minimizing the variance of v with respect to k , the residual r = σ 2 22 − σ 12 σ 2 11 + σ 2 22 − 2 σ 12 r 1 + σ 2 11 − σ 12 σ 2 11 + σ 2 22 − 2 σ 12 r 2 = 1 2 r 1 + 1 2 r 2 = f 2 + v is obtained for k = 1 / 2 , with variance σ 2 v = k 2 σ 2 11 + 2 k (1 − k ) σ 12 + (1 − k ) 2 σ 2 22 = 1 . Comparing this value with the v ariance of the noise in the individual residuals r 1 and r 2 , i.e., σ 2 11 = σ 2 22 = 3 , shows that by using the larger testable PSO set M 1 = { e 1 , e 2 , e 3 , e 5 } it is possible to construct a residual with a higher fault-to-noise ratio than for either r 1 or r 2 alone. The technique used above to combine the two residuals is a standard example from the theory of sensor fusion [11]. Higher order of redundancy can be exploited in many dynamic and nonlinear models to further suppress model and measurement noise when designing residuals. In the MSO algorithm in [14] and the MTES algorithm in [13], a crucial step is the computation of the o verdetermined part M + defined as the largest subset of M that is a PSO set. The operator M ∗ , defined below , will replace the operator M + in the algorithm introduced in the Section IV. The definition of the operator M + was extened in [18] to an operator that giv es the largest PSO subset of a model for which all variables can be computed under the assumption that linear square subsystems can be in verted and subject to causality constraints between variables. In [6], the monitorable part of a model was defined in an analogous way under deriv ativ e, integral, and mixed causality assumptions. The following general operator will be considered in this paper . Definition 4 (The operator M ∗ ) . Given a set of equations M , the set M ∗ is defined as the larg est testable PSO subset of M . C. F ault signatur e Different residuals are sensitiv e to different sets of faults, and the possible sets of fault signatures of the residuals are giv en by the follo wing definition. Definition 5 (Fault signature) . Let F ( M ) denote the set of faults included in a model M . A set of faults F  = ∅ such that F = F ( M ) for some testable PSO set M is called a fault signatur e. The set M 1 = { e 1 , e 2 , e 3 , e 5 } used above has the fault signature F ( M 1 ) = { f 2 } , and M 1 is the largest testable PSO set with this fault signature. Hence, the fault-to-noise ratio cannot be improved further by adding more equations without increasing the set of faults which the residual is sensitiv e to. This makes M 1 suitable for constructing a residual sensitive to f 2 . The following lemma shows that for any given fault signa- ture there is a largest testable PSO set with that signature. Lemma 1. Given a fault signature F , ther e exists a unique testable PSO set M such that F ( M ) = F and M ′ ⊂ M for all testable PSO sets M ′ satisfying F ( M ′ ) = F . Pr oof. Let M be the union of all testable PSO sets M ′ such that F ( M ′ ) = F . By Definition 5, there e xists at least one testable PSO set M ′ with F ( M ′ ) = F , so the union is non- empty . Moreover , M is a testable PSO set since it is a union of testable sets, and it clearly contains e very such M ′ as a subset. It follows that the sets gi ven by Lemma 1 are of particular interest when designing residuals, which motiv ates the follo w- ing definition. Definition 6 (RG set) . A testable PSO set M is called a Residual Generation set (RG set) if it is the lar gest testable PSO set such that F ( M ) = F for a given fault signatur e F . 4 D. F ault detectability and fault isolability The concepts structurally detectable faults and structurally isolability of multiple faults from [13] are now modified by replacing the M + operator by M ∗ . It is assumed that a fault f affects only one equation and that equation is denoted by e f . Definition 7 (Structurally detectable) . Let F ( M ) denote the set of faults included in a model M . A fault f is structurally detectable in M if f ∈ F ( M ∗ ) . A fault mode is represented by a set F i of faults, and a fault mode F i is isolable from a fault mode F j if there exists a residual sensitiv e to some fault of F i but insensitive to all faults in F j . A formal definition is: Definition 8 (Structurally isolable) . A fault mode F i is struc- turally isolable fr om mode F j in a model M if F i ∩ ( M \ eq ( F j )) ∗  = ∅ , wher e eq ( F j ) = ∪ f ∈ F j { e f } . The set of all TESs forms a partially ordered set under set inclusion, where the MTESs are the minimal elements in the graph-theoretical sense. For each TES there is an associated set of faults included in the corresponding submodel giv en by the follo wing definition from [13]. Definition 9 (TS and MTS) . A subset of faults F is a test support TS if there exists a PSO set M such that F ( M ) = F . A test support is a minimal test support MTS if no pr oper submodel is a test support There is a one-to-one correspondence between the TESs and the TSs, and the collection of all TSs characterizes the complete multiple-fault isolability properties of the model. Hence, the MTESs and the corresponding MTSs contain all information about the multiple fault isolability properties of the model. W e now turn to the corresponding results for RG sets. In general, the set of all RG sets also forms a partially ordered set under inclusion. There is again a one-to-one correspon- dence between the RG sets and all possible fault signatures, which completely characterizes the multiple-fault isolability properties of the model under the restrictions considered in this paper . The main difference between TESs and RG sets can be illustrated using model (1). In this example, there are two RG sets, M 1 = { e 1 , e 2 , e 3 , e 5 } and M 2 = { e 1 , e 2 , e 3 , e 4 , e 5 } , with corresponding fault signatures F 1 = { f 2 } and F 2 = { f 1 , f 2 } , respectiv ely . The set M 1 is the only minimal element. Consequently , it is not sufficient to consider only the mini- mal RG sets when analyzing the structural fault detection and isolation properties of the model. A union of RG sets is a subset of an RG set with a fault signature that is equal to the union of the fault signatures of the individual RG sets. Hence, the union of RG sets does not provide any additional information about the fault isolability properties of the model. This moti vates the follo wing definition. Definition 10 (Irreducible fault signature and IRG set) . A fault signatur e is called irreducible if it cannot be written as a union other fault signatur es. An RG set is called an Irreducible RG (IRG) set if its fault signatur e is irr educible. In graph-theoretical terms, the set of all fault signatures can be viewed as a join-semilattice, where the join operation is gi ven by set union, and the irreducible fault signatures of the join-irreducible elements in this structure; see, for example, [2]. The follo wing result summarizes the discussion and shows that, in the context of this paper , IRG sets play a role analogous to that of MTESs. Theorem 1. Ther e is a one-to-one corr espondence between the set of all RG sets and the set of all fault signatur es. Furthermor e, the fault signature of any RG set can be written as as a union of irreducible fault signatur es, and the set of all irreducible fault signatur es is the minimal set with this pr operty . Pr oof. The one-to-one correspondence follows directly from Definition 6 and Lemma 1. By Definition 10, any fault signature is either an irreducible fault signatur itself, or can be represented as a union of irreducible fault signatures. Finally , any collection of sets with this property must contain all irreducible fault signatures, which proves minimality . I V . A L G O R I T H M The operator M ∗ will be used in the algorithm belo w . Before presenting it, three important differences compared to the MTES algorithm are highlighted, since they must be taken into account when modifying the MTES algorithm. The first difference concerns a key operation. In the MTES algorithm, a frequently used operation is to remove one equa- tion and then compute the overdetermined part, ( M \ { e } ) + . If e is an arbitrary equation in a PSO set M , then φ  ( M \ { e } ) +  = φ ( M ) − 1 , that is, the order of redundancy decreases by one. In the algorithm belo w , the corresponding operation is to first remov e one equation and then compute the largest testable subset ( M \ { e } ) ∗ . For system (1), the order of redundancy is φ ( M ) = 3 . If equation e 5 is remov ed, the largest testable subset is empty , ( M \ { e 5 } ) ∗ = ∅ , and thus φ (( M \ { e 5 } ) ∗ ) = 0 < φ ( M ) − 1 = 2 . The second difference is that, for a giv en model, all MTESs hav e the same order of redundancy , which can be computed in adv ance; see Lemma 3 in [13]. This property is used as a stopping criterion in the MTES algorithm. In contrast, the IRG sets of model (1) hav e different orders of redundanc y: φ ( { e 1 , e 2 , e 3 , e 5 } ) = 2 and φ ( { e 1 , e 2 , e 3 , e 4 , e 5 } ) = 3 . The third difference is related to structural isolability as defined in [12], which is symmetric: if f i is isolable from f j , then f j is isolable from f i . In the example in Section III, the two fault signatures are { f 2 } and { f 1 , f 2 } . This means that f 2 is structurally isolable from f 1 , but not vice versa. W ithout this symmetry , it is not possible to define equiv alence classes as in Section V .B of [14], which were used to improv e the efficienc y of the MSO and MTES algorithms. Furthermore, 5 due to the lack of symmetry , it is not possible to define the sets R which were used in these algorithms to prev ent that the same set was found more than once. The algorithm belo w follo ws the same basic ideas as in [13]. It is initialized with the PSO sets M = M ∗ 0 , where M 0 is the model to be analyzed. Equations with faults are remov ed and largest testable subsets are computed alternately until the there is a set without faults reached in each branch of the recursion tree. The output of the algorithm is the set of all RG subsets of M 0 . It is straightforward to identify which RG sets are IRG sets using Definition 10, so this step is not included here. 1 function S = FindRG ( M ) 2 S = ∅ ; 3 while F ( R )  = ∅ 4 Select an e ∈ R such that F ( { e } )  = ∅ 5 M ′ = ( M \ { e } ) ∗ ; 6 if F ( M ′ )  = ∅ 7 S = S ∪ { M ′ } ; 8 S = S ∪ FindRG ( M ′ ); 9 end 10 end 11 end One modification compared to [13] is that the operator M + has been replaced by M ∗ in line 5. Moreover , since the order of redundancy of the IRG sets is not known a priori, the stopping criterion in line 6 is that F ( M ′ ) = F (( M \ { e } ) ∗ ) be- comes empty , rather than reaching a precomputed redundancy lev el. This situation occurs, for example, if e 5 is removed from system (1). The section concludes with a proof establishing the correct- ness of the algorithm. Theorem 2. The output of the algorithm is the set of all R G sets in the model M 0 . Pr oof. Input to the algorithm is M = M ∗ 0 . W e show that an arbitrary RG set M ′ ⊊ M is found at least once by the algorithm by constructing a branch in the recursion tree from M to M ′ . Assume that f ∈ F ( M ) \ F ( M ′ ) , and define M s = ( M \ { e f } ) ∗ . W e claim that F ( M ′ ) ⊂ F ( M s ) . If this were not the case, then F ( M s ) ⊊ F ( M ′ ∪ M s ) ⊂ F ( M ) \ { f } = F ( M \ { e f } ) , where M ′ ∪ M s ⊂ M is a testable PSO set, which contradicts the definition of M s . If M s = M ′ , the claim follo ws. Otherwise, we repeat the procedure abov e with M replaced by M s until M ′ is reached. V . E X A M P L E : D I FF E R E N T I A L A L G E B R A I C E Q U A T I O N S In this section, semi-explicit linear dif ferential-algebraic equations (D AEs) of the form ˙ x 1 = A 11 x 1 + A 12 x 2 + B 1 u , (2a) 0 = A 21 x 1 + A 22 x 2 + B 2 u , (2b) y = C 1 x 1 + C 2 x 2 , (2c) are be used to illustrate how the theory de veloped in the previous sections can be applied to identify subsystems that are compatible with a specified residual generation methodology . The follo wing example is considered. e 1 : ˙ x 1 = x 2 − x 1 e 2 : ˙ x 2 = x 1 + x 2 − 2 x 3 + f 1 e 3 : ˙ x 3 = x 2 − x 3 e 4 : y 1 = x 1 + f 2 e 5 : y 2 = x 3 + f 3 e 6 : y 3 = x 1 − x 3 (3) which is an ODE system. Howe ver , the analysis below con- siders subsystems of (3), which are in general semi-explicit linear D AEs. In this example, a set of equations is defined to be struc- turally testable if it has low structural differential index. The differential index can be seen as a measure of how far a D AE is from being an ordinary differential equation (ODE); an index of zero corresponds to an ODE system. Models with index zero or one are referred to as lo w-index D AEs and can be integrated using standard ODE solvers; see, e.g., [1]. The number of unkno wn signals is equal to the number of equation in the subsystem (2a), and (2b). Such a quadratic system is a lo w-index D AE system if A 22 is in vertible. In this linear case (2b) giv es x 2 = − A − 1 22 ( A 21 x 1 + B 2 u ) and substitution into (2a) yields an ODE system. In [7], the concept of lo w index was extended to a structural property of non-quadratic, structurally overdetermined PSO systems, and model (2) has low structural index if and only if the matrix  A 22 C 2  has full structural column rank. It was sho wn ho w such struc- turally ov erdetermined, lo w-index submodels can be used for residual generation for more general non-linear DAE systems, either by direct sequential residual generation with integral causality or observer based residual methods. In Section IV .B of [6], an operator was introduced that returns the largest PSO set that is structurally monitorable under integral causality , and a semi-explicit DAE has low structural index if and only if it is structurally monitorable under integral causality; see Theorem 14 in [7]. The submodel M ∗ is defined as largest PSO subset of M that is structurally monitorable under integral causality . Applying the algorithm in Section IV with the operator M ∗ defined above yields four IRG sets, and the corresponding irreducible fault signatures, as shown in T able I. 6 IRG set Fault signature { e 1 , e 2 , e 3 , e 6 } { f 1 } { e 1 , e 2 , e 3 , e 4 , e 6 } { f 1 , f 2 } { e 1 , e 2 , e 3 , e 5 , e 6 } { f 1 , f 3 } { e 4 , e 5 , e 6 } { f 2 , f 3 } T ABLE I I R G S E T S A ND FAU LT S IG N A T U R ES O F T H E M O DE L (3) T able II lists the MTESs and their corresponding test supports for the model. The follo wing observations can be made. First, { e 1 , e 2 , e 3 , e 9 } is the only IRG set that is also an MTES. Second, the faults { f 1 , f 5 , f 6 } are mutually isolable in the MTES framework, in the sense that for each fault there exists a fault signature that contains this fault but not the others, whereas under the computational constraint only f 1 is isolable from f 5 and f 6 . This follows from the fact that the MTESs { e 1 , e 3 , e 4 , e 6 } and { e 1 , e 3 , e 5 , e 6 } are not lo w-index D AEs. The reason for this is that the variable x 2 cannot be determined algebraically in either of the two cases. Howe ver , by adding the equation e 2 to these sets, the second and third IRG sets are obtained, at the cost of increased fault sensitivity due to the inclusion of the fault f 1 . Finally , it can be noted that the fault mode { f 5 , f 6 } is isolable from { f 1 } in the IRG framew ork. MTES T est support { e 1 , e 2 , e 3 , e 6 } { f 1 } { e 1 , e 3 , e 4 , e 6 } { f 2 } { e 1 , e 3 , e 5 , e 6 } { f 3 } T ABLE II M T ES S A N D T ES T S U P PO RT S O F T H E M O D E L (3) V I . C O N C L U S I O N S The paper has introduced a generalized structural frame work for residual generation that simultaneously accounts for fault locations and constraints imposed by a chosen residual genera- tion methodology . By extending the notions of PSO sets, MSO sets, and TES/MTES to testable PSO sets and RG sets, and by characterizing irreducible fault signature sets as the join- irreducible elements of the resulting lattice, the framew ork captures both fault isolability and computational limitations in a unified way . An algorithm for computing all RG sets has been proposed, obtained by replacing the o verdetermined-part operator with an operator M ∗ that encodes the admissible residual generation method. The D AE example demonstrates how structural dif fer- ential index can be used to define M ∗ and to deriv e IRG sets and their associated irreducible fault signatures. The results show that IRG sets play , for method-constrained diagnosis, the same central role that MTESs play in previous works. R E F E R E N C E S [1] Uri M Ascher and Linda R Petzold. Computer methods for ordinary differ ential equations and differential-alg ebraic equations , volume 61. Siam, 1998. [2] Garrett Birkhoff. Lattice theory . Third edition. American Mathematical Society Colloquium Publications, V ol. XXV . American Mathematical Society , Providence, R.I., 1967. [3] Mogens Blanke, Michel Kinnaert, Jan Lunze, and Marcel Staroswiecki. Diagnosis and fault-tolerant contr ol . Springer, 3 edition, 2016. [4] A. L. Dulmage and N. S. Mendelsohn. Coverings of bipartite graphs. Canadian Journal of Mathematics , 10:517–534, 1958. [5] William C Dunn. Fundamentals of industrial instrumentation and pr ocess control . 2005. [6] Erik Frisk, Anibal Bregon, Jan ˚ Aslund, Mattias Krysander, Belarmino Pulido, and Gautam Biswas. Diagnosability analysis considering causal interpretations for differential constraints. IEEE T ransactions on Sys- tems, Man, and Cybernetics – P art A: Systems and Humans , 42(5):1216– 1229, September 2012. [7] Erik Frisk, Mattias Krysander, and Jan ˚ Aslund. Analysis and design of diagnosis systems based on the structural differential index. In IF AC W orld Congress , T oulouse, France, 2017. [8] Erik Frisk, Mattias Krysander, and Daniel Jung. A toolbox for analysis and design of model based diagnosis systems for large scale mod- els. IF AC-P apersOnLine , 50(1):3287–3293, 2017. 20th IF AC W orld Congress. [9] E.R. Gelso, S.M. Castillo, and J. Armengol. An algorithm based on structural analysis for model-based fault diagnosis. In Artificial Intel- ligence Resear ch and Development. F r ontiers in Artificial Intelligence and Applications , volume 184, pages 138–147. IOS Press, 2008. [10] Maxence Glotin, Louise Trav ´ e-Massuy ` es, and Elodie Chanthery . MSO Sets and MTES for Dummies. In 35th International Conference on Principles of Diagnosis and Resilient Systems (DX 2024) , volume 125 of Open Access Series in Informatics (O ASIcs) , pages 13:1–13:15, Dagstuhl, Germany , 2024. [11] Steven M. Kay . Fundamentals of Statistical Signal Processing , V olume I: Estimation Theory . Prentice Hall, Englewood Cliffs, NJ, USA, 1993. [12] M. Krysander and E. Frisk. Sensor placement for fault diagnosis. IEEE T ransactions on Systems, Man, and Cybernetics – P art A: Systems and Humans , 38(6):1398–1410, 2008. [13] Mattias Krysander, Jan ˚ Aslund, and Erik Frisk. A structural algorithm for finding testable sub-models and multiple fault isolability analysis. 21st International W orkshop on Principles of Diagnosis (DX-10), Port- land, Oregon, USA, 2010. [14] Mattias Krysander, Jan ˚ Aslund, and Mattias Nyberg. An efficient algorithm for finding minimal over -constrained sub-systems for model- based diagnosis. IEEE T ransactions on Systems, Man, and Cybernetics – P art A: Systems and Humans , 38(1), 2008. [15] J. Armengol Llobet, A. Bregon, T . Escobet, E. R. Gelso, M. Krysander, M. Nyberg, X. Olive, B. Pulido, and L. T rave-Massuyes. Minimal structurally overdetermined sets for residual generation: A comparison of alternati ve approaches. In Proceedings of IF AC Safepr ocess’09 , Barcelona, Spain, 2009. [16] Sebastian Proell, Fabian Jarmolowitz, and Jan Lunze. A comprehensive observer -based fault isolation method with application to a hydraulic power train. In 8th IF A C Symposium on Advances in Automotive Contr ol , K olm ˚ arden, Sweden, 2016. [17] B. Pulido and C. Alonso-Gonz ´ alez. Possible Conflicts: a compilation technique for consistency-based diagnosis. IEEE T rans. on Systems, Man, and Cybernetics. P art B: Cybernetics , 34(5):2192–2206, Octubre 2004. [18] Albert Rosich, Erik Frisk, Jan ˚ Aslund, Ramon Sarrate, and Fatiha Nej- jari. Fault diagnosis based on causal computations. IEEE T ransactions on Systems, Man, and Cybernetics – P art A: Systems and Humans , 42(2):371–381, 2012. [19] Carl Sv ¨ ard and Mattias Nyberg. Residual generators for fault diagnosis using computation sequences with mixed causality applied to automoti ve systems. IEEE T ransactions on Systems, Man, and Cybernetics – P art A: Systems and Humans , 40(6):1310–1328, 2010. [20] L. Tra ve-Massuyes, T . Escobet, and X. Olive. Diagnosability analysis based on component-supported analytical redundancy relations. IEEE T ransaction on Systems, Man, and Cybernetics – P art A , 36(6):1146– 1160, 2006.