Dulmage-Mendelsohn Decomposition Research Articles

Dilation Choice Sets, Dulmage–Mendelsohn Decomposition, and Structural Controllability

Dilation Choice Sets, Dulmage–Mendelsohn Decomposition, and Structural Controllability

Real-Time Multi-Sensor Joint Fault Diagnosis Method for Permanent Magnet Traction Drive Systems Based on Structural Analysis.

Sensor faults are one of the most common faults that cause performance degradation or functional loss in permanent magnet traction drive systems (PMTDSs). To quickly diagnose faulty sensors, this paper proposes a real-time joint diagnosis method for multi-sensor faults based on structural analysis. Firstly, based on limited monitoring signals on board, a structured model of the system was established using the structural analysis method. The isolation and detectability of faulty sensors were analyzed using the Dulmage-Mendelsohn decomposition method. Secondly, the minimum collision set method was used to calculate the minimum overdetermined equation set, transforming the higher-order system model into multiple related subsystem models, thereby reducing modeling complexity and facilitating system implementation. Next, residual vectors were constructed based on multiple subsystem models, and fault detection and isolation strategies were designed using the correlation between each subsystem model and the relevant sensors. The validation results of the physical testing platform based on online fault data recordings showed that the proposed method could achieve rapid fault detection and the localization of multi-sensor faults in PMTDS and had a good application value.

Minimal Controllability Problem on Linear Structural Descriptor Systems With Forbidden Nodes

We consider a minimal controllability problem (MCP), which determines the minimum number of input nodes for a descriptor system to be structurally controllable. We investigate the “forbidden nodes” in descriptor systems, denoting nodes that are unable to establish connections with input components. The three main results of this work are as follows. First, we show a solvability condition for the MCP with forbidden nodes using graph theory such as a bipartite graph and its Dulmage–Mendelsohn decomposition. Next, we derive the optimal value of the MCP with forbidden nodes. The optimal value is determined by an optimal solution for constrained maximum matching, and this result includes that of the standard MCP in the previous work. Finally, we provide an efficient algorithm for solving the MCP with forbidden nodes based on an alternating path algorithm.

Fault Diagnosability Analysis of Multi-Mode Systems

Multi-mode systems can operate in different modes, leading to large numbers of different dynamics. Consequently, applying traditional structural diagnostics to such systems is often untractable. To address this challenge, we present a multi-mode diagnostics algorithm that relies on a multi-mode extension of the Dulmage-Mendelsohn decomposition. We introduce two methodologies for modeling faults, either as signals or as Boolean variables, and apply them to a modular switched battery system in order to demonstrate their effectiveness and discuss their respective advantages.

Finding Hall Blockers by Matrix Scaling

Given a nonnegative matrix [Formula: see text], the matrix scaling problem asks whether A can be scaled to a doubly stochastic matrix [Formula: see text] for some positive diagonal matrices D1, D2. The Sinkhorn algorithm is a simple iterative algorithm, which repeats row-normalization [Formula: see text] and column-normalization [Formula: see text] alternatively. By this algorithm, A converges to a doubly stochastic matrix in limit if and only if the bipartite graph associated with A has a perfect matching. This property can decide the existence of a perfect matching in a given bipartite graph G, which is identified with the 0, 1-matrix AG. Linial et al. (2000) showed that [Formula: see text] iterations for AG decide whether G has a perfect matching. Here, n is the number of vertices in one of the color classes of G. In this paper, we show an extension of this result. If G has no perfect matching, then a polynomial number of the Sinkhorn iterations identifies a Hall blocker—a vertex subset X having neighbors [Formula: see text] with [Formula: see text], which is a certificate of the nonexistence of a perfect matching. Specifically, we show that [Formula: see text] iterations can identify one Hall blocker and that further polynomial iterations can also identify all parametric Hall blockers X of maximizing [Formula: see text] for [Formula: see text]. The former result is based on an interpretation of the Sinkhorn algorithm as alternating minimization for geometric programming. The latter is on an interpretation as alternating minimization for Kullback–Leibler (KL) divergence and on its limiting behavior for a nonscalable matrix. We also relate the Sinkhorn limit with parametric network flow, principal partition of polymatroids, and the Dulmage–Mendelsohn decomposition of a bipartite graph. Funding: K. Hayashi was supported by the Japan Society for the Promotion of Science [Grant JP19J22605]. H. Hirai was supported by Precursory Research for Embryonic Science and Technology [Grant JPMJPR192A].

Applications of the Dulmage–Mendelsohn decomposition for debugging nonlinear optimization problems

Nonlinear modeling and optimization is a valuable tool for aiding decisions by engineering practitioners, but programming an optimization problem based on a complex electrical, mechanical, or chemical process is a time-consuming and error-prone activity. Therefore, there is a need for model analysis and debugging tools that can detect and diagnose modeling errors. One such tool is the Dulmage–Mendelsohn decomposition, which identifies structurally under- and over-determined subsets in systems of equations and variables by partitioning the bipartite graph of the system. This work provides the necessary background to understand the Dulmage–Mendelsohn decomposition and its application to the analysis of nonlinear optimization problems, demonstrates its use in diagnosing a variety of modeling errors, and introduces software implementations for analyzing nonlinear optimization problems in the Pyomo and JuMP algebraic modeling languages.

Model-Based Fault Analysis and Diagnosis of PEM Fuel Cell Control System

This paper presents a systematic fault analysis and diagnosis method of a PEM fuel cell control system using a model-based approach. With a model-based approach, it is possible to analyze the causal relationship and effect of probable faults in the system, and to diagnose them under the assumption that the model and the process are similar. With a model-based approach, it is possible to analyze the causal relationship and effect of probable faults in the system and diagnose them under the assumption that the model and the process are similar. In this work, a model-based approach was adopted for fault analysis and diagnosis, and its methods are suggested. A PEM fuel cell is mathematically modelled, analyzed, and verified for the analysis and simulations. Relationships among variables are shown using an incidence matrix and with a Dulmage–Mendelsohn decomposition of the matrix. When it is difficult to detect faults due to a deficient degree of redundancy, a bi-partite graph is used to analyze the effect of faults and to assess the possibility of fault detection through the appropriate redundant sensor placement. Thereafter, residuals are obtained based on analytical redundancies of the system, and a fault signature matrix is subsequently constructed. A fault detection and isolation (FDI) algorithm is developed based on a fault signature matrix that describes the connection between faults and residuals. The simulation results demonstrate the validity and effectiveness of the proposed FDI algorithm for diagnosing faults. With the proposed FDI algorithm, eight faults could be diagnosed by FDI algorithm with given sensors in the system.

Graft Analogue of general Kotzig–Lovász decomposition

Canonical decompositions, such as the Gallai–Edmonds and Dulmage–Mendelsohn decompositions, are a series of structure theorems that form the foundation of matching theory. The classical Kotzig–Lovász decomposition is a canonical decomposition applicable to a special class of graphs with perfect matchings, called factor-connected graphs, and is famous for its contribution to the studies of perfect matching polytopes and lattices. Recently, this decomposition has been generalized for general graphs with perfect matchings; this generalized decomposition is called the general Kotzig–Lovász decomposition . In fact, this generalized decomposition can be presented as a component of a more composite canonical decomposition called the Basilica decomposition . As such, the general Kotzig–Lovász decomposition has contributed to the derivation of various new results in matching theory, such as an alternative proof of the tight cut lemma and a characterization of maximal barriers in general graphs. Joins in grafts, also known as T -joins in graphs, is a classical concept in combinatorics and is a generalization of perfect matchings in terms of parity. More precisely, minimum joins and grafts are generalizations of perfect matchings and graphs with perfect matchings, respectively. Under the analogy between matchings and joins, analogues of the canonical decompositions for grafts are expected to be strong and fundamental tools for studying joins. In this paper, we provide a generalization of the general Kotzig–Lovász decomposition for arbitrary grafts. Our result also contains Sebö’s generalization of the classical Kotzig–Lovász decomposition for factor-connected grafts. From our results in this paper, a generalization of the Dulmage–Mendelsohn decomposition, which is originally a classical canonical decomposition for bipartite graphs, has been obtained for bipartite grafts. This paper is the first of a series of papers that establish a generalization of the Basilica decomposition for grafts.

Real-Time Detection of Incipient Inter-Turn Short Circuit and Sensor Faults in Permanent Magnet Synchronous Motor Drives Based on Generalized Likelihood Ratio Test and Structural Analysis.

This paper presents a robust model-based technique to detect multiple faults in permanent magnet synchronous motors (PMSMs), namely inter-turn short circuit (ITSC) and encoder faults. The proposed model is based on a structural analysis, which uses the dynamic mathematical model of a PMSM in an frame to evaluate the system’s structural model in matrix form. The just-determined and over-determined parts of the system are separated by a Dulmage–Mendelsohn decomposition tool. Subsequently, the analytical redundant relations obtained using the over-determined part of the system are used to form smaller redundant testable sub-models based on the number of defined fault terms. Furthermore, four structured residuals are designed based on the acquired redundant sub-models to detect measurement faults in the encoder and ITSC faults, which are applied in different levels of each phase winding. The effectiveness of the proposed detection method is validated by an in-house test setup of an inverter-fed PMSM, where ITSC and encoder faults are applied to the system in different time intervals using controllable relays. Finally, a statistical detector, namely a generalized likelihood ratio test algorithm, is implemented in the decision-making diagnostic system resulting in the ability to detect ITSC faults as small as one single short-circuited turn out of 102, i.e., when less than 1% of the PMSM phase winding is short-circuited.

Model-Based Fault Tolerant Control for the Thermal Management of PEMFC Systems

Proton exchange membrane fuel cells (PEMFCs) are considered as one of the most promising power sources for transportation in the near future. Proper and robust thermal management is a key issue in fuel cell applications. This paper mainly studies the sensor fault detection and isolation (FDI) and fault tolerant control (FTC) for the thermal management of PEMFC systems. The thermal model of fuel cell is established and analyzed by structure analysis, and the residual generator of sensor faults are designed by Dulmage–Mendelsohn decomposition to rearrange the biadjacency matrix of the thermal model. A sliding-mode-based active FTC (AFTC) strategy is proposed for the thermal management of fuel cell systems. The effectivenesses of the proposed FDI method and AFTC strategy are verified on a fuel cell test bench. The experimental results show that the temperature of the PEMFC stack can be maintained at the reference value with high accuracy even when sensor fails.

Fault Isolability Analysis and Optimal Sensor Placement for Fault Diagnosis in Smart Buildings

Faults and anomalies in buildings are among the main causes of building energy waste and occupant discomfort. An effective automatic fault detection and diagnosis (FDD) process in buildings can therefore save a significant amount of energy and improve the comfort level. Fault diagnosability analysis and an optimal FDD-oriented sensor placement are prerequisites for effective, efficient and successful diagnostics. This paper addresses the problem of fault diagnosability for smart buildings. The method used in the paper is a model-based technique which uses Dulmage-Mendelsohn decomposition. To the best of our knowledge, this is the first time that this method is used for applications in smart buildings. First a dynamic model for a zone in a real-case building is developed in which faults are also introduced. Then fault diagnosability is investigated by analyzing the fault isolability of the model. Based on the investigation, it was concluded that not all the faults in the model are diagnosable. Then an approach for placing new sensors is implemented. It is observed that for two test scenarios, placing additional sensors in the model leads to full diagnosability. Since sensors placement is key for an effective FDD process, the optimal placement of such sensors is also studied in this work. A case study of campus building OU44 at the University of Southern Denmark is considered. The results show that as the system gets more complicated by introducing more faults, additional sensors should be added to achieve full diagnosability.

Model-Based Fault Diagnosis of an Anti-Lock Braking System via Structural Analysis

The anti-lock braking system (ABS) is an essential part in ensuring safe driving in vehicles. The Security of onboard safety systems is very important. In order to monitor the functions of ABS and avoid any malfunction, a model-based methodology with respect to structural analysis is employed in this paper to achieve an efficient fault detection and identification (FDI) system design. The analysis involves five essential steps of SA applied to ABS, which includes critical faults analysis, fault modelling, fault detectability analysis and fault isolability analysis, Minimal Structural Over-determined (MSO) sets selection, and MSO-based residual design. In terms of the four faults in the ABS, they are evaluated to be detectable through performing a structural representation and making the Dulmage-Mendelsohn decomposition with respect to the fault modelling, and then they are proved to be isolable based on the fault isolability matrix via SA. After that, four corresponding residuals are generated directly by a series of suggested equation combinations resulting from four MSO sets. The results generated by numerical simulations show that the proposed FDI system can detect and isolate all the injected faults, which is consistent with the theoretical analysis by SA, and also eventually validated by experimental testing on the vehicle (EcoCAR2) ABS.

Decomposition theorems for square-free [formula omitted]-matchings in bipartite graphs

A Ck-free 2-matching in an undirected graph is a simple 2-matching which does not contain cycles of length k or less. The complexity of finding the maximum Ck-free 2-matching in a given graph varies depending on k and the type of input graph. The case where k=4 and the graph is bipartite, which is called the maximum square-free 2-matching problem in bipartite graphs, is well-solved. Previous results for this special case include min-max theorems, polynomial combinatorial algorithms, a linear programming formulation with dual integrality for a weighted version, and discrete convex structures.In this paper, we further investigate the structure of square-free 2-matchings in bipartite graphs and present new decomposition theorems. These theorems serve as analogues of the Dulmage-Mendelsohn decomposition for matchings in bipartite graphs and the Edmonds-Gallai decomposition for matchings in nonbipartite graphs. We exhibit two canonical minimizers for the set function in the min-max formula, and a characterization of the maximum square-free 2-matchings with the aid of these canonical minimizers.

On the fixed controllable subspace in linear structured systems

In this paper we consider interconnected networks that are described by means of structured linear systems with state and control variables. We represent these systems, whose matrices contain fixed zeros and free parameters, by means of directed graphs and study questions concerning controllability and the controllable subspace.We show in this paper that the controllable subspace can have a part that will be present for almost all values of the free parameters. It actually is a subspace of the controllable subspace and will be referred to as the fixed controllable subspace. The subspace can then be seen as a kind of robustly controllable part of the system. Indeed, it is a subspace in the state space with the generic property that states in it can be steered in an arbitrary way.We derive a characterization of the fixed controllable subspace using the graph representation. The obtained characterization makes use of well-known algorithms from optimization and networks theory. To get some more insight in the components in the fixed part, we also give a representation of the structured linear systems by means of bipartite graphs. Using the Dulmage–Mendelsohn decomposition, we are able to decompose our structured systems in such a way that in some special cases, the fixed controllable subspace can be obtained directly from the decomposition.

Constrained Min-Cut Replication for K-Way Hypergraph Partitioning

Replication is a widely-used technique in information retrieval and database systems for providing fault tolerance and reducing parallelization and processing costs. Combinatorial models based on hypergraph partitioning are proposed for various problems arising in information retrieval and database systems. We consider the possibility of using vertex replication to improve the quality of hypergraph partitioning. In this study, we focus on the constrained min-cut replication (CMCR) problem, where we are initially given a maximum replication capacity and a K-way hypergraph partition with an initial imbalance ratio. The objective in the CMCR problem is finding the optimal vertex replication sets for each part of the given partition such that the initial cut size of the partition is minimized, where the initial imbalance is either preserved or reduced under the given replication capacity constraint. In this study, we present a complexity analysis of the CMCR problem and propose a model based on a unique blend of coarsening and integer linear programming (ILP) schemes. This coarsening algorithm is derived from a novel utilization of the Dulmage-Mendelsohn decomposition. Experiments show that the ILP formulation coupled with the Dulmage-Mendelsohn decomposition-based coarsening provides high quality results in practical execution times for reducing the cut size of a given K-way hypergraph partition.

Problem formulation and analysis for optimal energy management in multisources systems: W effect

PurposeThe purpose of this paper is to introduce a problematic phenomenon that can occur when managing multi electrical sources systems by optimization.Design/methodology/approachThe energy management problem is formulated as a linear optimisation problem. Two approaches are developed and applied to detect the possible existence of equivalents solutions. The first is based on Dulmage‐Mendelsohn (DM) decomposition. With this method the structure of the optimisation problem is analysed. The second approach is a numeric approach; the detection of equivalents solutions is made by the formulation of new optimisation problem and the objective function of this problem is to maximise the distance between two equivalents solutions.FindingsThe numeric approach is more efficient than the structural approach. In some cases, applying DM decomposition may not be sufficient to detect the risk of W effect. This is because DM decomposition does not take the value of variable's coefficient into consideration, which is important to determine the degrees of freedom in the set of variables.Originality/valueMulti sources systems are widely used, especially in buildings where renewable energies have good potential application. The linear formulation of the management problem may induce an existence of equivalent command strategies. The detection approach presented in this paper shows that some solutions are better than others from an applicabability point of view. They will not exhaust rapidly the storage system. This approach can be implemented in virtual sources plant to avoid solutions with this so‐called W effect.

Efficient Optimal Sensor Placement for Structural Model Based Diagnosis

This work aims to study which sensors are required to be installed in a process in order to improve certain fault diagnosis specifications. Especially, the present method is based on structural models. Thus, system models involving a wide variety of equations (e.g. linear, non-linear algebraic, dynamics) can be easy handled. The use of structural models permits to define the diagnosis properties from the Dulmage-Mendelsohn decomposition, avoiding in this way the computation of any minimal redundant subsystem. Furthermore, in the present paper, the cost of the sensor configuration is considered. Therefore the proposed method attempts to find not all the possible solution but the optimal one. The optimal search is efficiently performed by developing an algorithm based on heuristic rules which, in general, allow to significantly reduce the search.

Matching structure of symmetric bipartite graphs and a generalization of Pólya's problem

A bipartite graph is said to be symmetric if it has symmetry of reflecting two vertex sets. This paper investigates matching structure of symmetric bipartite graphs. We first apply the Dulmage–Mendelsohn decomposition to a symmetric bipartite graph. The resulting components, which are matching-covered, turn out to have symmetry. We then decompose a matching-covered bipartite graph via an ear decomposition, which is a sequence of subgraphs obtained by adding an odd-length path repeatedly. We show that, if a matching-covered bipartite graph is symmetric, an ear decomposition can retain symmetry by adding no more than two paths. As an application of these decompositions to combinatorial matrix theory, we present a natural generalization of Pólya's problem. We introduce the problem of deciding whether a rectangular { 0 , 1 } -matrix has a signing that is totally sign-nonsingular or not, where a rectangular matrix is totally sign-nonsingular if the sign of the determinant of each submatrix with the entire row set is uniquely determined by the signs of the nonzero entries. We show that this problem can be solved in polynomial time with the aid of the matching structure of symmetric bipartite graphs. In addition, we provide a characterization of this problem in terms of excluded minors.

Bilateral bargaining in networks

Each connected pair of nodes in a network can jointly produce one unit of surplus. A maximum number of linked nodes is selected in every period to bargain bilaterally over the division of the surplus, according to the protocol proposed by Rubinstein and Wollinsky [Equilibrium in a market with sequential bargaining, Econometrica 53 (1985) 1133–1150]. All pairs, which reach an agreement, obtain the (discounted) payoffs and are removed from the network. This bargaining game has a unique subgame perfect equilibrium that induces the Dulmage–Mendelsohn decomposition (partition) of the bipartite network (of the set of nodes in this network).

A Graph Theoretical Approach to a Software Structure Analysis Applying Dulmage-Mendelsohn Decomposition to a Function-Data Digraph

A Graph Theoretical Approach to a Software Structure Analysis Applying Dulmage-Mendelsohn Decomposition to a Function-Data Digraph