Fault indicator

If banks are not rotating correctly there can be entrapment of air in the material—and this will give rise to marks in the surface of the film known in the trade by many names, including tick marks and crow s feet . The variety of different names for such faults indicates just how common they are. 

Defects in CVD diamond were studied in Ref. [102]. Figure 7.8 shows a presence of a mirror plane (indicated by M) in a crystal, and Figure 7.9 includes both mirror planes and 111 stacking faults (indicated by SF). These data suggest that CVD diamond in general contains numerous defects in the atomic scale. 

In this paper, the proposed approach to fault detection and isolation is a model-based approach. The first part of this communication focuses on the main fundamental concepts of the simulation library PrODHyS, which allows the simulation of the system reference model of a typical process example. Then, the proposed detection approach is presented. This exploits the extended Kalman Filter, in order to generate a fault indicator. In the last part, this approach is exploited through the simulation of the monitoring of a didactic example. 

The last part deals with the diagnosis of the fault (hatehed pattern in the Figure 1). The signature obtained in the previous part is eompared with the theoretieal fault signatures by means of distance. A theoretical signature T.j of a partieular default j is obtained by experience or in our case, by simulations of the proeess with different oeeureney dates of this fault. Then, a fault indicator is generated. For this, we define two distanees the... 

The geological evidence that has been recorded along the Lanterman and Leap Year faults indicates that both thrusting and strike-slip motions have occurred. Therefore, two kinds of tectonic models have been proposed ... 

The SRO s decision was a mistake since the operators did not increase safely injection (SI) to compensate for the low SCM [subcooUng margin], this could be labelled an error of omission (EOO). However, the operators apparently reduced SI according to the EOP [Emergency Operating Procedures] and the faulted indicator. Hence, the error could he termed an EOC. ... 

In online model-based FDI, ARRs are evaluated using measurements from the real system being subject to disturbances. The time evolution of ARR residuals serve as fault indicators. For hybrid systems, ARRs are system mode dependent. Hence, an unobserved mode change invalidates the actual set of ARRs. As a result, computed values of fault indicators may exceed current thresholds indicating faults in some system components that have not happened. ARR residuals derived from a bond graph can not only serve as fault indicators but may also be used for model-based system mode identification. 

Chapter4 presents bond graph model-based quantitative FDI for hybrid systems. As the approach uses ARRs and their residuals as fault indicators, the generation of... 

As fault indicators should be sensitive to real faults but insensitive to parameter uncertainties, adaptive system mode dependent thresholds are needed for FDI in hybrid systems robust with regard to parameter uncertainties. Chapter 5 demonstrates that incremental bond graph can serve this purpose for switched LTI systems. To that end some basics of incremental bond graphs are recalled. It is shown how parameter sensitivities of ARRs and ARR thresholds can be obtained. A small example illustrates the approach. 

Chapter demonstrates that ARR residuals cannot only serve as fault indicators but may also be used for system mode identification in online FDI. First, the general case of switched LTI systems is considered and it is assumed that ARRs can be expressed in explicit form relating known system inputs and inputs either obtained by measurements or by simulation of the real system behaviour. A small example illustrates bond graph based system mode identification using ARRs. 

Once a quantitative system model is available, different methods can be used for the generation of a fault indicator or a residual as a primary step in FDI. These methods are either based on observers or a bank of observers [26, 27], on parity relations [28, 29], or on analytical redundancy relations [30, 31], or on parameter estimation [32, 33]. In case a fault has occurred in the system, the time evolution of some residuals must deviate distinctly from that during normal healthy system operation. 

Consideration of the equations of a faulty LTI system and of a Luenberger state variable observer reveals that any faults affecting the system have an affect on the observer output error which, after transients have settled, can be used as a fault indicator ([34], Sect. 5.2.2). Assume that thedynamics of a system may be represented by the linear time-invariant state space model... 

The output estimation error Cy(t) is a good candidate for a residual as it depends on the faults f t) and the disturbances d t) but not on the input values u t). In the case of no faults and no disturbances this fault indicator apparently vanishes for t oo. 

Finally, changes in the physical system parameter can serve as fault indicators. The procedure is displayed in Fig. 1.2. 

A single set of model equations as well as equations for fault indicators valid for all system modes can be (automatically) derived from the bond graph. They contain the discrete moduli tm (t) e 0,1, i = 1,..., of the switching MTFs. Inserting values for the moduli belonging to a feasible switch state combination gives the equations that hold for the associated system mode. 

The representation of a hybrid system model by means of a bond graph with system mode independent causalities has the advantage that a unique set of equations can be derived from the bond graph that holds for all system modes. Discrete switch state variables in these equations account for the system modes. In this chapter, this bond graph representation is used to derive analytical redundancy relations (ARRs) from the bond graph. The result of their numerical evaluation called residuals can serve as fault indicator. Analysis of the structure of ARRs reveals which system components, sensors, actuators or controllers contribute to a residual if faults in these devices happen. This information is usually expressed in a so-called structural fault signature matrix (FSM). As ARRs derived from the bond graph of a hybrid system model contain discrete switch state variables, the entries in a FSM are mode dependent. Moreover, the FSM is used to decide if a fault has occurred and whether it can unequivocally be attributed to a component. Finally, the chapter discusses the numerical computation of ARRs. 

An essential step in FDI is the evaluation of the time history of residuals serving as fault indicators. To that end, one approach is to establish ARRs from a model of a system (cf. Fig. 1.3). These relations are algebraic or dynamic constraints between known continuous variables, i.e. system inputs u and measured output variables y that include known model parameters O. ARRs derived from a bond graph of a hybrid system model also depend on discrete switch state variables. For a healthy system, the time evolution of ARRs should ideally be identical to zero in all system modes. In practice, residuals will be within certain small error bounds due to measurement noise, parameter uncertainties, and numerical inaccuracies. If, however, faults in some system components occur, then the values of some residuals will be outside given thresholds and can serve as fault indicators. The sfructure of ARRs expressed in the FSM will then indicate whether faults can be isolated. Let m t) denote the vector... 

The measured variables ji obtained from the faulty system model and the outputs yi of the non-faulty system model may be coupled by feeding the differences y,- - y,-into modulated sinks that deliver an output r, so that their input becomes zero. These sinks are termed residual sinks. The output r, of a residual sink is a power variable and is inserted into the balance of power variables at that junction in the non-faulty system model from which its co-variable y, is obtained. The outputs of the residual sinks are additional inputs into the non-faulty system model that force the faultless system to alter its behaviour so that it becomes identical to the one of the faulty system. If no fault is introduced into the real system model, then there are no differences and all values r, (f) are close to zero. Differences, however, lead to values r, (r) that remain distinguishably different from zero as long as a fault is effective, i.e. is not repaired. That is, the outputs of the residual sinks can serve as fault indicators. The balance of power variables at a junctions in the non-faulty system model connected to a residual sink becomes an ARR when unknowns have been eliminated and the output of the residual sink becomes the residual of that ARR. 

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. 

ARR residuals as fault indicators should be distinctly sensible to true faults and robust with regard to parameter uncertainties. That is, if parameters varies, the time evolution of ARR residuals should be within prescribed bounds. For real systems described by a hybrid model bounds should be adapted to system modes as the dynamic behaviour can be quite different in different system modes. 

As to EDI, mode-dependent ARRs can be deduced from a DBG with storage elements in derivative causality and sensors in inverted causality. Their structure is captured in an all-mode ESM. An evaluation of ARRs yields residuals that are used as fault indicators. In general, ARRs relate time derivatives of known variables. The necessary differentiation is carried out in discrete time. 

ARR residuals serving as fault indicators should be distinguishably sensitive to true faults but little sensitive to noise and parameter faults in order to avoid false alarms on the one hand side and to make sure that fault detection does not miss any faults. Therefore, appropriate thresholds for ARR residuals are to be set. As the dynamic behaviour of hybrid systems can be quite different in different modes, predefined bounds of constant value may not be suitable. In this book, the incremental bond graph approach [4] has been briefly recalled and applied to hybrid system models to deduce adaptive mode-dependent ARR residual thresholds that account for parameter uncertainties. 

Definition A.12 Residual) Fault indicator based on deviations between measurements and model equation based calculations. 

Analytical redundancy relations (ARRs) Are mathematical equations that relate known system inputs, known parameters and quantities obtained by measurements from a real system. Their evaluation results in so-called ARR residuals that are identical to zero or close to zero in narrow limits as long as the system is healthy. Residuals that deviate distinguishably from zero serve as fault indicators. If nonlinear constitutive element equations do not permit to eliminate unknown variables in a candidate for an ARR in closed symbolic form then residuals are given implicitly and can be determined by numerically solving a set of equations. As inputs into ARRs may be time derivatives of measured quantities, measurement noise is to be filtered appropriately. The differentiation is carried out in discrete time. 

Fault isolation If values of fault indicators are beyond acceptable limits and an alarm has been raised then fault isolation means to locate possibly faulty components. 

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