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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.

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.