Diagnostic bond graph

In Fig. 1.3, the real system includes actuators and sensors. The ARRs have been obtained from a model of the system that may include models of sensors and acmators depending on whether possible faults in these components are considered or not. In the FDI block, ARR residuals are used to detect and to isolate faults. As known inputs and measurements from the real system are inputs into the ARRs, their evaluation must take place in real-time. The ARRs, however, can be generated off-line by deducing them from a so-called diagnostic bond graph model as indicated in Fig. 1.4. 

Derivation of ARRs from a diagnostic bond graph then starts by summing power variables at all those junctions that have a BG sensor element in inverted causality attached to it. At first, these balances of power variables will contain unknown variables. They may be eliminated by following causal paths and by using constitutive equations of bond graph elements. The result may be a set of ARRs in closed symbolic form [cf. (4.2)] if nonlinear constitutive element equations permit necessary... 

Figure4.2 shows a diagnostic bond graph (DBG) of the switched network in Fig.4.1 with the semiconductor switch explicitly modelled by an element Sw b, i.e. an MTF-R pair.

Fig. 4.2 Invariant causality diagnostic bond graph of the switched network in Fig. 4.1...

The structural information contained in the ARRs, i.e. the information on which ARR depends on which component parameters can be obtained directly by inspection of causal paths in a diagnostic bond graph [1]. There is no need to derive equations and to eliminate unknowns in order to set up a mode-dependent FSM. To that end, causal paths from model inputs to inputs of sensor elements are considered. Elements that are traversed on these causal paths contribute to the ARR of a residual related to a sensor element. An output of a source or an element that is followed directly or indirectly by switches on the causal path to a sensor element provides an entry in the FSM equal to the product of the switch states. 

In the same manner, mode-dependent ARRs can be generated from diagnostic bond graphs of hybrid system models with more than one switch. As an example. Fig. 4.6 depicts the diagnostic bond graph of the switched network in Fig. 3.4. 

In order to isolate the faulty component the coherence vector is matched with the rows of the FSM, i.e with the component fault signatures. Given a hybrid system model, there is a FSM for each system mode. That is, in order to use the correct FSM for comparison, it is important to know in which mode the monitored system is at the present time point. Chapter shows that ARRs derived from a diagnostic bond graph can also be used for system mode identification. 

Coupling of a Behavioural and a Diagnostic Bond Graph Model... 

As to the numerical computation of ARR residuals, two possible bond graph based approaches have been presented. One approach suited for online as well as for offline FDI is to use a diagnostic bond graph in which storage elements are in preferred derivative causality in order to be independent of initial conditions that are difficult to be obtained in online FDI. Moreover, sensors are in inverted causality. 

For an offline FDI, the real system may be replaced by a behavioural model that allows to introduce deliberately all kinds of fault and to study their effect with no risk. The equations derived from a model of the faulty system are integrated with respect to time by means of an appropriate solver starting from initial conditions, while the derivatives of inputs into the diagnostic bond graph model are computed in discrete time. Evaluation of the faulty system model and computation of the ARR residuals can be performed by means of Scilab script files. 

Samantaray, A. K., Medjaher, K., Quid Bouamama, B., Staroswiecki, M., Dauphin-Tanguy, G. (2006). Diagnostic bond graphs for online fault detection and isolation. Simulation Modelling Practice and Theory, 14 3), 237-262. 

As a result, ARR residuals as fault indicators may be obtained by evaluating ARRs derived from a diagnostic bond graph with nominal parameters. In order to assess the effect of uncertain parameters on ARR residuals, parameter variations of ARR residuals may be derived from an incremental bond graph. Application of the triangle inequality then gives adaptive bounds for these variations. 

Figure 5.11 shows a diagnostic bond graph of the simple switched RC-circuit. For this circuit, ARRs (4.6)-(4.7) simplily to... 

Fig. 5.11 Diagnostic bond graph of the simple switched RC-circuit...

Figure5.15 illustrates this situation assuming that measurement uncertainties are additive. A flow / = f + Af with a predicted part / and an uncertain part A/ due to measurement uncertainty is the output of a non-ideal sensor and an input into the diagnostic bond graph. The input / into the diagnostic bond graph results in an effort e = e + Ae that controls a modulated sink MSe where e denotes the predicted part and Ae the uncertain part. The output w = 5(e - - Ae )A0 of the modulated sink is an input into the incremental bond graph that is needed to compute the variation Ar of an ARR residual r.

If measurement uncertainties can be assumed to be bounded, then application of the triangle inequality may yield thresholds for parameter variations of ARR residuals that are independent of measurement uncertainties. For instance, let z be the predicted part of an output variable z of the diagnostic bond graph that controls a modulated sink of the incremental bond and let Az < be the bounded measurement uncertainty. Furthermore, let Ar be the variation of an ARR residual r that depends on z and its derivative. Then... 

ARRs may be obtained as outputs of a diagnostic bond graph. If unknowns in ARR candidates can be eliminated, they are functions in closed form of known inputs u t), known measurements y(t), known system parameters 0 and the system mode denoted by all discrete switch states m t). Let r denote the /th residual, then... 

Sensitivities of the outputs of a model with respect to a parameter can be derived from a sensitivity bond graph [5-7]. Sensitivities of ARR residuals with respect to a parameter can be obtained from incremental bond graphs (Chap. 5), from sensitivity pseudo bond graphs [8] and from diagnostic sensitivity bond graphs [9]. 

In the example, the gradient of the least squares cost function with respect to the target fault parameters has been computed numerically. For the simple network, ARRs in closed symbolic form can be derived from the diagnostic bond graph so that partial derivatives of the cost function with respect to parameters can be provided in symbolic form. 

As to the previously considered simple hybrid network (Fig. 4.1), ARRs in closed symbolic form could be derived from its diagnostic bond graph (Fig. 4.2). As a result, analytical expressions for the ARR residuals to be used in the formulation of the least squares cost function are available and symbolic partial derivatives of ARR residuals with respect to the targeted fault parameters can be exploited. 

If parameters are to be estimated in real-time, only present measured values and past values sampled over a time window of finite length stored in an array are available. Nevertheless, they are inputs into the diagnostic model and solving its equations yields values for the ARR residuals. As a diagnostic bond graph model used for... 

Fig. 8.14 Incremental diagnostic bond graph of the boost converter in Fig. 8.1...

In the sequel, the uncontrolled three phase rectifier in Fig. 8.36 is considered. It is straightforward to convert its circuit schematic into a bond graph. Figures.38 displays a diagnostic bond graph with flow sensors Df ia, Df and Df ic for the line currents and an effort sensor De Ud for the load voltage. 

Fig. 8.38 Diagnostic bond graph of the three-phase rectifier in Fig. 8.36...

The components contributing to the ARR residuals can also be identified directly on the diagnostic bond graph in Fig. 8.38 by following causal paths from a sensor element in inverse causality into the bond graph and back to it. For instance, there is a causal path from and back to the flow detector Df ia. 

---