Yet More Elementary Proof of Matrix-Tree Theorem for Signed Graphs

A signed graph G˙=(G,σ)is a graph G=(V(G),E(G))with vertex set V(G)and edge set E(G),together with a functionσ:E→{+1,−1}assigning a positive or negative sign to each edge.In this paper,we present a more elementary proof for the matrix-tree theorem of signed graphs,which is based on the relations between the incidence matrices and the Laplcians of signed graphs.As an application,we also obtain the results of Monfared and Mallik about the matrix-tree theorem of graphs for signless Laplacians.

Bounds for the Least Laplacian Eigenvalue of a Signed Graph

A signed graph is a graph with a sign attached to each edge. This paper extends some fundamental concepts of the Laplacian matrices from graphs to signed graphs. In particular, the relationships between the least Laplacian eigenvalue and the unbalancedness of a signed graph are investigated.

Some Properties of a Class of Addition Cayley Signed Graph

Let G=Cay^(+)(Z_(p1)×Z_(p1)^(α1)_(p2)^(α2)…_(pk)^(αk),Φ)be a simple graph having vertex set V(G)=Z_(p1)×Z_(p1)^(α1)_(p2)^(α2)…_(pk)^(αk)and edge set E(G)={{x,y}:x+y∈Φ},where all p_(1),p_(2),…,p_(k)are distinct prime factors andΦis the set of all units of the ring Z_(p1)^(α1)_(p2)^(α2)…_(pk)^(αk).LetΣ=(G,σ)be a signed graph whose underlying graph is G and signature function isσ:E(G)→{+1,-1}defined asσ({x,y})={+1,ifx∈φ(p_(1))or y∈φ(p_(1)^(α1)p_(2)^(α2)…p_(k)^(αk));-1,otherwise.In this paper,we characterize the balance ofΣand some graphs derived from it such asη(Σ),L(Σ)and C_(E)(Σ).Moreover,we investigate the clusterability and sign-compativility ofΣ.

Signed Directed Graph and Qualitative Trend Analysis Based Fault Diagnosis in Chemical Industry

In the past 30 years,signed directed graph(SDG) ,one of the qualitative simulation technologies,has been widely applied for chemical fault diagnosis.However,SDG based fault diagnosis,as any other qualitative method,has poor diagnostic resolution.In this paper,a new method that combines SDG with qualitative trend analysis(QTA) is presented to improve the resolution.In the method,a bidirectional inference algorithm based on assumption and verification is used to find all the possible fault causes and their corresponding consistent paths in the SDG model.Then an improved QTA algorithm is used to extract and analyze the trends of nodes on the consis-tent paths found in the previous step.New consistency rules based on qualitative trends are used to find the real causes from the candidate causes.The resolution can be improved.This method combines the completeness feature of SDG with the good diagnostic resolution feature of QTA.The implementation of SDG-QTA based fault diagno-sis is done using the integrated SDG modeling,inference and post-processing software platform.Its application is illustrated on an atmospheric distillation tower unit of a simulation platform.The result shows its good applicability and efficiency.

Sensor Configuration and Test for Fault Diagnoses of Subway Braking System Based on Signed Digraph Method

Fault diagnosis of various systems on rolling stock has drawn the attention of many researchers. However, obtaining an optimized sensor set of these systems, which is a prerequisite for fault diagnosis, remains a major challenge. Available literature suggests that the configuration of sensors in these systems is presently dependent on the knowledge and engineering experiences of designers, which may lead to insufficient or redundant development of various sensors. In this paper, the optimization of sensor sets is addressed by using the signed digraph （SDG） method. The method is modified for use in braking systems by the introduction of an effect-function method to replace the traditional quantitative methods. Two criteria are adopted to evaluate the capability of the sensor sets, namely, observability and resolution. The sensors configuration method of braking system is proposed. It consists of generating bipartite graphs from SDG models and then solving the set cover problem using a greedy algorithm. To demonstrate the improvement, the sensor configuration of the HP2008 braking system is investigated and fault diagnosis on a test bench is performed. The test results show that SDG algorithm can improve single-fault resolution from 6 faults to 10 faults, and with additional four brake cylinder pressure （BCP） sensors it can cover up to 67 double faults which were not considered by traditional fault diagnosis system. SDG methods are suitable for reducing redundant sensors and that the sensor sets thereby obtained are capable of detecting typical faults, such as the failure of a release valve. This study investigates the formal extension of the SDG method to the sensor configuration of braking system, as well as the adaptation supported by the effect-function method.

Qualitative Algebra and Graph Theory Methods for Dynamic Trend Analysis of Continuous System

Qualitative algebraic equations are the basis of qualitative simulation,which are used to express the dynamic behavior of steady-state continuous processes.When the values and operation of qualitative variables are redefined,qualitative algebraic equations can be transformed into signed direct graphs,which are frequently used to predict the trend of dynamic changes.However,it is difficult to use traditional qualitative algebra methods based on artificial trial and error to solve a complex problem for dynamic trends.An important aspect of modern qualitative algebra is to model and characterize complex systems with the corresponding computer-aided automatic reasoning.In this study,a qualitative affection equation based on multiple conditions is proposed,which enables the signed di-rect graphs to describe complex systems better and improves the fault diagnosis resolution.The application to an industrial case shows that the method performs well.

Practical bipartite consensus for multi-agent systems:A barrier function-based adaptive sliding-mode control approach

This paper is concerned with bipartite consensus tracking for multi-agent systems with unknown disturbances.A barrier function-based adaptive sliding-mode control(SMC)approach is proposed such that the bipartite steady-state error is converged to a predefined region of zero in finite time.Specifically,based on an error auxiliary taking neighboring antagonistic interactions into account,an SMC law is designed with an adaptive gain.The gain can switch to a positive semi-definite barrier function to ensure that the error auxiliary is constrained to a predefined neighborhood of zero,which in turn guarantees practical bipartite consensus tracking.A distinguished feature of the proposed controller is its independence on the bound of disturbances,while the input chattering phenomenon is alleviated.Finally,a numerical example is provided to verify the effectiveness of the proposed controller.

SDG-based Model Validation in Chemical Process Simulation

Signed direct graph (SDG) theory provides algorithms and methods that can be applied directly to chemical process modeling and analysis to validate simulation models, and is a basis for the development of a software environment that can automate the validation activity. This paper is concentrated on the pretreatment of the model validation. We use the validation scenarios and standard sequences generated by well-established SDG model to validate the trends fitted from the simulation model. The results are helpful to find potential problems, assess possible bugs in the simulation model and solve the problem effectively. A case study on a simulation model of boiler is presented to demonstrate the effectiveness of this method.

Probabilistic SDG model description and fault inference for large-scale complex systems

Large-scale complex systems have the feature of including large amount of variables that have complex relationships, for which signed directed graph （SDG） model could serve as a significant tool by describing the causal relationships among variables. Although qualitative SDG expresses the causing effects between variables easily and clearly, it has many disadvantages or limitations. Probabilistic SDG proposed in the article describes deliver relationships among faults and variables by conditional probabilities, which contains more information and performs more applicability. The article introduces the concepts and con- struction approaches of probabilistic SDG, and presents the inference approaches aiming at fault diagnosis in this framework, i.e. Bayesian inference with graph elimination or junction tree algorithms to compute fault probabilities. Finally, the probabilistic SDG of a typical example of 65t/h boiler system is given.

Fault diagnosis of AMT gear shifting process based on semi-quantitative SDG model

In order to diagnose gear shifting process in automated manual transmission（AMT）,a semi-quantitative signed directed graph（SDG）model is applied.Mathematical models are built by analysis of the power train dynamic and the gear shifting control process.The SDG model is built based on related priori knowledge.By calculating the fuzzy membership degree of each compatible passway and its possible fault source,we get the possibilities of failure for each possible fault source.We begin with the nodes with the maximum possibility of failure in order to find the failed part.The diagnosis example shows that it is feasible to use the semi-quantitative SDG model for fault diagnosis of the gear shifting process in AMT.

Nowhere-zero 15-Flow in 3-Edge-connected Bidirected Graphs

It was conjectured by Bouchet that every bidirected graph which admits a nowhere-zero κ flow will admit a nowhere-zero 6-flow. He proved that the conjecture is true when 6 is replaced by 216. Zyka improved the result with 6 replaced by 30. Xu and Zhang showed that the conjecture is true for 6-edge-connected graphs. And for 4-edge-connected graphs, Raspaud and Zhu proved it is true with 6 replaced by 4. In this paper, we show that Bouchet＇s conjecture is true with 6 replaced by 15 for 3-edge-connected graphs.

Synchronization of second-order Kuramoto networks from the perspective of edge dynamics

This paper presents new synchronization conditions for second-order phase-coupled Kuramoto oscillators in terms of edge dynamics.Two types of network-underlying graphs are studied,the positively weighted and signed graphs,respectively.We apply an edge Laplacian matrix for a positively weighted network to represent the edge connections.The properties of the edge Laplacian matrix are analyzed and incorporated into the proposed conditions.These conditions take account of the dynamics of edge-connected oscillators instead of all oscillator pairs in conventional studies.For a network with positive and negative weights,we represent the network by its spanning tree dynamics,and derive conditions to evaluate the synchronization state of this network.These conditions show that if all edge weights in the spanning tree are positive,and the tree-induced dynamics are in a dominant position over the negative edge dynamics,then this network achieves synchronization.The theoretical findings are validated by numerical examples.

SDG-Based HAZOP and Fault Diagnosis Analysis to the Inversion of Synthetic Ammonia

This paper presents some practical applications of signed directed graphs （SDGs） to computeraided hazard and operability study （HAZOP） and fault diagnosis, based on an analysis of the SDG theory. The SDG is modeled for the inversion of synthetic ammonia, which is highly dangerous in process industry, and HAZOP and fault diagnosis based on the SDG model are presented. A new reasoning method, whereby inverse inference is combined with forward inference, is presented to implement SDG fault diagnosis based on a breadth-first algorithm with consistency rules. Compared with conventional inference engines, this new method can better avoid qualitative spuriousness and combination explosion, and can deal with unobservable nodes in SDGs more effectively. Experimental results show the validity and advantages of the new SDG method.

Neural Modified Extended State Observer for Autonomous Maritime Transportation Vehicles with Actuator Faults and Unmatched Nonlinearities

This paper investigates extended state observer design problem for autonomous maritime transportation vehicles consisting of multiple unmanned aerial vehicles(UAVs)and unmanned surface vehicles(USVs)via the cooperation–competition network.Through two coordinate transformations,the dynamic model of UAVs and USVs is transformed into a standard secondorder heterogeneous multi-agent system in unified earth-fixed frame.In order to recover the unavailable velocities,unpredictable fault,total disturbance as well as to estimate the heterogeneous match/unmatched nonlinearities,neural network(NN)is incorporated into the design process of the modified distributed extended state observer(MESO).Based on the Lyapunov stability analysis,it is proved that all the error signals are uniformly ultimately bounded(UUB)and the bounds could be arbitrarily small by choosing appropriate parameters.Simulation results verify the effectiveness of the proposed method.