Numerical Computation of ARR Residuals

ARRs from a bond graph representing a hybrid system model and the condensation of structural information in a system mode dependent stmctural fault signature matrix are presented. The chapter concludes by addressing off-line numerical computation of ARR residuals. To that end, a model of a system subject to faults is coupled to a bond graph of the healthy system by means of so-called residual sinks. This way, faults may be deliberately introduced without any risk and their effect, detection and isolation may be studied. 

In some way, this coupling of two models by residual sinks may be compared to the temporary coupling of two bodies such as the plates of a clutch by a residual sink (cf. Fig. 2.15). As long as the clutch is disengaged, there is no force acting between the two plates. If, however, the clutch engages, a torque acts on both of them such that the difference of their angular velocities is zero. The load side is forced to adapt to drive side. This approach has been applied for the numerical computation of ARR residuals from continuous time models [14, 17] and recently also to systems described by a hybrid model [18]. 

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. 

Fig. 8.27 Coupling of a faulty system model to a non-faulty system model for the numerical computation of ARR residuals...

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. 

Figure4.13 indicates the basic idea of this approach to a numerical off-line computation of ARR residuals.

As a result, the computational cost of a parameter optimisation may be much higher in comparison to the case in which residuals are computed from available measured or computed data points either by using numerical differentiation or by evaluating analytical expressions. Clearly, if two coupled models are to be evaluated in order to get numerical values of ARR residuals, the gradient of the least squares cost function with respect to the vector of the targeted fault parameters is to be determined numerically. All other parts of the parameter estimation procedure remain unchanged. 

For ARRs in closed symbolic form, parameter sensitivities of ARR residuals can be obtained by symbolic differentiation. In case an explicit formulation of ARRs is not achievable, e.g. due to nonlinear algebraic loops, parameter sensitivities of ARR residuals can be numerically computed by using a sensitivity bond graph, in which bonds carry sensitivities of power variables [12-14], or by using incremental bond graphs, in which bonds carry variations of power variables [5]. In Chap. 5, incremental bond graphs are used for the determination of adaptive fault thresholds. 

Essential advantages of this framework are that ARRs in symbolic form are not required. The diagnostic model is numerically computed by calculating derivatives of its inputs in discrete time. The result is the time evolution of ARR residuals. Finally, in the case of slow real processes, off-line simulation using a behavioural model reveals the effects of parametric faults much faster than a real-time computation of ARRs that uses measured output from the real process. 

If ARRs can be obtained in closed symbolic form, parameter sensitivities can be determined by symbolic differentiation with respect to parameters. If this is not possible, parameter sensitivities of ARRs can be computed numerically by using either a sensitivity bond graph [1 ] or an incremental bond graph [5, 6]. Incremental bond graphs were initially introduced for the purpose of frequency domain sensitivity analysis of LTI models. Furthermore, they have also proven useful for the determination of parameter sensitivities of state variables and output variables, transfer functions of the direct model as well as of the inverse model, and for the determination of ARR residuals from continuous time models [7, Chap. 4]. In this chapter, the incremental bond graph approach is applied to systems described by switched LTI systems. 

Multiple parameter fault isolation by means of minimisation of least squares of ARR residuals needs residual parameter sensitivity functions if a gradient search based method is used. If ARRs can be derived in closed symbolic form from a bond graph, their analytical expressions can be used in the formulation of the least squares cost function and can be differentiated with respect to the vector of targeted parameters either numerically or residuals as functions of the targeted parameters can be differentiated symbolically. If ARRs are not available in symbolic form, they can be numerically computed by solving the equations of a DBG. 

This system mode identification may require considerable computational costs especially if ARRs cannot be deduced in closed symbolic form so that the entire DBG model is to be evaluated to obtain numerically the time history of ARR residuals. However, as it is one and same the problem with different sets of discrete switch state variables, this computation can be easily and efficiently performed in parallel on multicore processors or multiprocessor computers. 

As ARRs in symbolic form cannot always be obtained by elimination of unknown variables, sensitivities of their residuals with respect to parameters sometimes cannot be derived by symbolic differentiation. Therefore, sensitivity bond graphs have been used for numerical computation of residual sensitivities [8, 11]. In the following, it is shown that once the matrices of the state space model have been derived from the original bond graph with nominal parameters and from the associated incremental bond graph, parameter sensitivities of residuals of ARRs can also be determined in symbolic form by multiplication of transfer matrix entries. 

Remark 4.2 This approach to the derivation of ARRs in closed form has been implemented in the software ModelBuilder [7]. Also, a module of the software Symbols [8] can automatically generate ARRs. If ARRs in closed form are not possible, i.e. unknowns cannot be eliminated then ARR residuals are to be computed numerically simultaneously with the model equations. ... 

Each of the multiple sensitivity BGs constructed from a DBG needs the same modulating signals as inputs which can be provided by the DBG. By coupling the multiple sensitivity BGs to the DBG, ARR residuals and their partial derivatives with respect to the parameters to be estimated can be numerically computed at the same time for each step of the parameter estimation procedure. This is indicated in Fig. 6.23. 

The ARR residuals have been computed numerically. To that end, a bond graph model of the faulty rectifier has been coupled to a bond graph of the healthy system by means of residual sinks as displayed in Fig. 8.45. 

ARR residuals can be numerically computed as unknowns of a mode-dependent DAE system. There is no need for ARRs in closed symbolic form. 

A vanishing phase iine current ia affects all three ARRs. Their residuals have been computed numerically by coupling a model of the faulty system to a model of the non-faulty system by means of residual effort sinks as depicted in Fig. 8.27. 

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