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For the case when the abstraction of instantaneous state transitions is adopted, this paper proposes to start fault detection and isolation in an engineering system from a single time-invariant causality bond graph representation of a hybrid model. To that end, the paper picks up on a long-known proposal to model switching devices by a transformer modulated by a Boolean variable and a resistor in fixed conductance causality accounting for its ON resistance. Bond graph representations of hybrid system models developed in this way have been used so far mainly for the purpose of simulation. The paper shows that they can well constitute an approach to the bond-graph-based quantitative fault detection and isolation of hybrid models. Advantages are that the standard sequential causality assignment procedure can be a used without modification. A single set of analytical redundancy relations valid for all physically feasible system modes can be (automatically) derived from the bond graph. Stiff model equations due to small values of the ON resistance in the switch model may be avoided by symbolic reformulation of equations and letting the ON resistance of some switches tend to zero, turning them into ideal switches.
First, for two examples considered in the literature, it is shown that the approach proposed in this paper can produce the same analytical redundancy relations as were obtained from a hybrid bond graph with controlled junctions and the use of a sequential causality assignment procedure especially for fault detection and isolation purpose. Moreover, the usefulness of the proposed approach is illustrated in two case studies by its application to standard switching circuits extensively used in power electronic systems and by simulation of some fault scenarios. The approach, however, is not confined to the fault detection and isolation of such systems. Analytically validated simulation results obtained by means of the program Scilab give confidence in the approach.
A bond graph representation of switching devices known for a long time has been a modulated transformer with a modulus b(t)∈{0,1}∀t≥0 in conjunction with a resistor R:Ron accounting for the ON-resistance of a switch considered non-ideal. Besides other representations, this simple model has been used in bond graphs for simulation of the dynamic behaviour of hybrid systems. A previous article of the author has proposed to use the transformer–resistor pair in bond graphs for fault diagnosis in hybrid systems. Advantages are a unique bond graph for all system modes, the application of the unmodified standard Sequential Causality Assignment Procedure, fixed computational causalities and the derivation of analytical redundancy relations incorporating ‘Boolean’ transformer moduli so that they hold for all system modes. Switches temporarily connect and disconnect model parts. As a result, some independent storage elements may temporarily become dependent, so that the number of state variables is not time-invariant. This article addresses this problem in the context of modelling and simulation of fault scenarios in hybrid systems. In order to keep time-invariant preferred integral causality at storage ports, residual sinks previously introduced by the author are used. When two storage elements become dependent at a switching time instance ts, a residual sink is activated. It enforces that the outputs of two dependent storage elements become immediately equal by imposing the conjugate3 power variable of appropriate value on their inputs. The approach is illustrated by the bond graph modelling and simulation of some fault scenarios in a standard three-phase switched power inverter supplying power into an RL-load in a delta configuration. A well-developed approach to model-based fault detection and isolation is to evaluate the residual of analytical redundancy relations. In this article, analytical redundancy relation residuals have been computed numerically by coupling a bond graph of the faulty system to one of the non-faulty systems by means of residual sinks. The presented approach is not confined to power electronic systems but can be used for hybrid systems in other domains as well. In further work, the RL-load may be replaced by a bond graph model of an alternating current motor in order to study the effect of switch failures in the power inverter on to the dynamic behaviour of the motor.