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In model-based fault detection and isolation (FDI), Analytical Redundancy Relations (ARRs) play a key role. Residuals as the result of their numerical evaluation serve as fault indicators. This paper proposes a novel approach to the generation of ARRs from a diagnostic bond graph (DBG) of a mode switching linear time invariant (LTI) model with ideal switches that hold for all modes of operation. Devices or phenomena with fast state transitions such as electronic diodes and transistors, clutches, or hard mechanical stops are modelled by ideal switches giving rise to variable causalities. Nevertheless, fixed causalities are assigned only once such that a DBG with storage elements in derivative causality and sensors in inverted causality is obtained. That is, the BG reflects the configuration for a specific system mode. From such a DBG with fixed causalities, a unique system of ARRs is derived from the DBG that holds for all system modes. The ARRs are implicitly given. In order to evaluate them, first, a set of algebraic or Differential Algebraic Equations (DAEs) must be solved. A formal matrix based approach that starts from the partitioning of a BG into fields is used for the general case. For illustration, two small system examples are considered. Their equations and the ARRs are directly derived from the DBG by following causal paths.

Integrating Bond Graph-Based Fault Diagnosis and Fault Accommodation Through Inverse Simulation
(2017)

This chapter addresses active fault tolerant control (FTC) of engineering systems represented by a mode switching linear time-invariant model. The presented approach integrates bond graph-based fault diagnosis and inverse simulation through solving a differential algebraic (DAE) system in order to reconstruct a system input after a severe fault has occurred.In this chapter, bond graph (BG) representations of hybrid models use switches. The standard causality assignment procedure (SCAP) is used to assign fixed causalities disregarding switch modes. Equations deduced from a BG are formulated in the declarative modelling language Modelica®; as a hybrid DAE system. Causality changes at switch ports are taken into account by the Modelica implementation of switches.As to fault detection, it is known that residuals of analytical redundancy relations (ARRs) deduced offline from a diagnostic bond graph (DBG) can serve as fault indicators. It is shown that they can also be used for estimating the magnitude of parametric faults in some simple cases.Moreover, ARR residuals can also be used in the reconstruction of a system input that compensates a severe fault. To that end, the forward model of the healthy system with nominal parameters derived from a BG is considered a DAE system of the inverse model. The output of the forward model of the healthy system in response to the initial known system input serving as the desired system output of the faulty system and the ARR residuals are inputs into the inverse model. Based on these inputs the DAE system then determines the input into the faulty system required for fault accommodation. As ARR residuals are used, fault isolation and estimation are not needed for input reconstruction. Alternatively, if isolation and estimation of the faulty parameter have been performed it can replace the nominal parameter in the inverse model and ARRs as inputs can be omitted.Computation of the forward model of the healthy system, the inverse model and the evaluation of the ARRs can be performed in parallel. Advantages of the presented approach based on ARRs and inverse simulation are that neither ARRs nor the reconstructed input are needed in closed analytical form. If constitutive equations of some elements do not permit an elimination of unknown variables, a DAE system deduced from a DBG has to be solved numerically in order to determine the ARR residuals used in the process of input reconstruction. The latter one also means to solve a DAE system numerically.

Initially, incremental bond graphs were introduced to support frequency domain sensitivity analysis of linearised time-invariant models. Subsequent publications have shown that they can be used for other purposes as well such as the determination of parameter sensitivities of analytical redundancy relations (ARRs) in symbolic form.

—This paper picks up on one of the ways reported in the literature to represent hybrid models of engineering systems by bond graphs with static causalities. The representation of a switching device by means of a modulated transformer (MTF) controlled by a Boolean variable in conjunction with a resistor has been used so far to build a model for simulation. In this paper, it is shown that it can also constitute an approach to bond graph based quantitative fault detection and isolation in hybrid system models. Advantages are that Analytical Redundancy Relations (ARRs) do not need to be derived again after a switch state has changed. ARRs obtained from the bond graph are valid for all system modes. Furthermore, no adaption of the standard sequential causality assignment procedure (SCAP) with respect to fault detection and isolation (FDI) is needed.

This paper proposes a novel approach to the generation of state equations from a bond graph (BG) of a mode switching linear time invariant model. Fast state transitions are modelled by ideal or non-ideal switches. Fixed causalities are assigned following the Standard Causality Assignment Procedure such that the number of storage elements in integral causality is maximised. A system of differential and algebraic equations (DAEs) is derived from the BG that holds for all system modes. It is distinguished between storage elements with mode independent causality and those that change causality due to switch state changes.

BGML - a novel XML format for the exchange and the reuse of bond graph models of engineering systems
(2006)

The paper proposes a novel XML based format called BGML that aims at supporting the exchange and the reuse of bond graph models of engineering systems between various bond graph and non-bond graph software. The validity of a BGML description of a bond graph model can be verified against an XML schema. Model equations and constitutive relations (linear or not) are represented in MathML. The concept of BGML is illustrated by means of a small example. In order to demonstrate the usefulness of BGML, an approach to transformations of formats used by bond graph software from and to XML and an export to non-bond graph software, as well as prototypes of their implementation in an experimental open source modelling and simulation environment are discussed.

The paper presents a bond graph model-based approach to active fault tolerant control (FTC) that makes use of residuals of analytical redundancy relations (ARRs). The latter ones are computed in order to decide whether a fault has occurred. Given a single fault hypothesis can be adopted, an advantage is that the time for isolating a fault among potential fault candidates that contribute to an ARR by means of parameter estimation may be saved and as long as ARR residuals are within their thresholds no input reconstruction at all is needed. It is shown that ARR residuals can be used for estimation of faults that can be isolated. ARR based input reconstruction is demonstrated by application to a buck-converter driven DC motor as a simple example of a switched power electronic system for which an averaged bond graph model is used. Scilab simulation runs confirm analytical results. If a required input cannot be determined analytically, it can be obtained by numerically solving a differential-algebraic equations (DAE) system.

Aiming at an automated bond graph based determination of unnormalized frequency domain sensitivities in symbolic form Borutzky and Granda proposed the systematic construction of a so-called incremental bond graph from an initial bond graph for the increments (Deltae)(t), (Deltaf)(t) of power variables e(t) andf(t) associated with each bond. This paper shows that the incremental bond graph can serve also as a starting point for setting up symbolically the canonical form as well as the standard interconnection form of state equations used for robustness study. The approach applicable to general linear time-invariant systems is illustrated by means of a fairly small example.

Multidisciplinary systems are described most suitably by bond graphs. In order to determine unnormalized frequency domain sensitivities in symbolic form, this paper proposes to construct in a systematic manner a bond graph from another bond graph, which is called the associated incremental bond graph in this paper. Contrary to other approaches reported in the literature the variables at the bonds of the incremental bond graph are not sensitivities but variations (incremental changes) in the power variables from their nominal values due to parameter changes. Thus their product is power. For linear elements their corresponding model in the incremental bond graph also has a linear characteristic. By deriving the system equations in symbolic state space form from the incremental bond graph in the same way as they are derived from the initial bond graph, the sensitivity matrix of the system can be set up in symbolic form. Its entries are transfer functions depending on the nominal parameter values and on the nominal states and the inputs of the original model. The sensitivities can be determined automatically by the bond graph preprocessor CAMP-G and the widely used program MATLAB together with the Symbolic Toolbox for symbolic mathematical calculation. No particular program is needed for the approach proposed. The initial bond graph model may be non-linear and may contain controlled sources and multiport elements. In that case the sensitivity model is linear time variant and must be solved in the time domain. The rationale and the generality of the proposed approach are presented. For illustration purposes a mechatronic example system, a load positioned by a constant-excitation d.c. motor, is presented and sensitivities are determined in symbolic form by means of CAMP-G/MATLAB.

Bond graph modelling was devised by Professor Paynter at the Massachusetts Institute of Technology in 1959 and subsequently developed into a methodology for modelling multidisciplinary systems at a time when nobody was speaking of object-oriented modelling. On the other hand, so-called object-oriented modelling has become increasingly popular during the last few years. By relating the characteristics of both approaches, it is shown that bond graph modelling, although much older, may be viewed as a special form of object-oriented modelling. For that purpose the new object-oriented modelling language Modelica is used as a working language which aims at supporting multiple formalisms. Although it turns out that bond graph models can be described rather easily, it is obvious that Modelica started from generalized networks and was not designed to support bond graphs. The description of bond graph models in Modelica is illustrated by means of a hydraulic drive. Since VHDL-AMS as an important language standardized and supported by IEEE has been extended to support also modelling of non-electrical systems, it is briefly investigated as to whether it can be used for description of bond graphs. It turns out that currently it does not seem to be suitable.

The square root characteristic commonly used to model the flow through hydraulic orifices may cause numerical problems because the derivative of the flow with respect to the pressure drop tends to infinity when the pressure drop approaches zero. Moreover, for small values of the pressure drop it is more reasonable to assume that the flow depends linearly on the pressure drop. The paper starts from an approximation of the measured characteristic of the discharge coefficient versus the square root of the Reynolds number given by Merritt and proposes a single empirical flow formula that provides a linear relation for small pressure differences and the conventional square root law for turbulent conditions. The transition from the laminar to the turbulent region is smooth. Since the slope of the characteristic is finite at zero pressure difference, numerical difficulties are avoided. The formula comprises physical meaningful terms and employs parameters which have a physical meaning. The proposed orifice model has been used in a bond graph model of a hydraulic sample circuit. Simulation results have proved to be accurate. The orifice model is easily implemented as a library model in a modern modeling language. Ultimately, the model can be adapted to approximate pipe flow losses as well.

In this paper we discuss an object oriented description of bond graph models of hydraulic components by means of the unified modeling language Modelica. A library which is still under development is briefly described and models of some standard hydraulic components are given for illustration. In particular we address the modeling of hydraulic orifices.