Fault indicators and unique mode-dependent state equations from a fixed-causality diagnostic bond graph of linear models with ideal switches

Abstract

Analytical redundancy relations are fundamental in model-based fault detection and isolation. Their numerical evaluation yields a residual that may serve as a fault indicator. Considering switching linear time-invariant system models that use ideal switches, it is shown that analytical redundancy relations can be systematically deduced from a diagnostic bond graph with fixed causalities that hold for all modes of operation. Moreover, as to a faultless system, the presented bond graph–based approach enables to deduce a unique implicit state equation with coefficients that are functions of the discrete switch states. Devices or phenomena with fast state transitions, for example, electronic diodes and transistors, clutches, or hard mechanical stops are often represented by ideal switches which give rise to variable causalities. However, in the presented approach, fixed causalities are assigned only once to a diagnostic bond graph. That is, causal strokes at switch ports in the diagnostic bond graph reflect only the switch-state configuration in a specific system mode. The actual discrete switch states are implicitly taken into account by the discrete values of the switch moduli. The presented approach starts from a diagnostic bond graph with fixed causalities and from a partitioning of the bond graph junction structure and systematically deduces a set of equations that determines the wanted residuals. Elimination steps result in analytical redundancy relations in which the states of the storage elements and the outputs of the ideal switches are unknowns. For the later two unknowns, the approach produces an implicit differential algebraic equations system. For illustration of the general matrix-based approach, an electromechanical system and two small electronic circuits are considered. Their equations are directly derived from a diagnostic bond graph by following causal paths and are reformulated so that they conform with the matrix equations obtained by the formal approach based on a partitioning of the bond graph junction structure. For one of the three mode-switching examples, a fault scenario has been simulated.