Parameter Sensitivities of ARR Residuals

the parameter sensitivity of ARR residual u with respect to the y th parameter 

The power variable zj is an output variable of the original bond graph model with nominal parameters and as such it is a weighted sum of the inputs Uk(t) into the original bond graph in the case of a switched LTI system. 

As a result, parameter sensitivities of ARR residuals r with respect to parameter can be obtained by constructing a matrix F from the matrices of the original 

These operations can be hardly manually performed, even for models of small size. However, a bond graph preprocessor such as CAMPG [11] can automatically derive the equations from the original as well as from the incremental bond graph. MATLAB [12] or Scilab [13] script files can then generate the matrices F and F in symbolic form and can perform the multiplication of a row of F by the factor F j8j for each requested parameter sensitivity of an ARR residual. 

For small switched LTI systems, variations of ARR residuals can be manually derived from an incremental bond graph by applying the principle of superposition. That is, only one bond graph element at a time is assumed to have an uncertain parameter. It is replaced by its incremental model. Detectors are replaced by a dual virtual detector for the variation of an ARR residual. Summing variations of flows or efforts, respectively, at these junctions and eliminating unknowns yields variations of residuals of ARRs as a weighted sum of the inputs supplied by those modulated sinks that represent parameter variations. The weighting factors in these sums are the sensitivities to be determined. 

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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. 

This chapter uses an incremental bond graph approach in order to determine parameter sensitivities of ARR residuals as well as adaptive mode-dependent ARR thresholds for systems described by a hybrid model. To that end, first, the underlying idea and some basics of incremental bond graphs are briefly recalled. 

In order to avoid that false alarms are reported to a supervisor system or that true faults are not detected, ARR residuals should be significantly sensitive to tme faults but little sensitive to parameter variations given uncertain system parameter values. Parameter sensitivities of ARR residuals can be singled out by defining appropriate thresholds. As the dynamic behaviour of a real system described by a hybrid model can be quite different in different system modes, thresholds should be adapted to system modes. 

Section 5.3.1 has shown that parameter sensitivities of ARR residuals may be obtained from an incremental bond graph. The latter bond graph can be systematically developed from an initial bond graph with nominal parameters by replacing elements with parameters to be estimated by their incremental component model. Inputs into the incBG are variations of the parameters to be estimated multiplied by a power variable of the initial BG. Outputs may be parameter variations of ARR residuals. They are a weighted sum of the parameter variations and the weighting factors are just the residual sensitivity functions. 

A sensitivity pseudo bond graph from which residual sensitivity functions for parameter estimation can be deduced is constructed by simply replacing those elements in a DBG by their sensitivity component model whose parameters are to be estimated. Equations for parameter sensitivities of ARR residuals can then be deduced from the SPBG in the same way as the equations of a state space model are deduced from a behavioural BG or equations for ARR variations from an incBG. 

In the illustrating example, residual sensitivity functions have been deduced from a single SPBG as partial derivatives of ARR residuals with respect to a non-specific parameter 0. The parameter sensitivities of ARR residuals needed for the parameter estimation procedure are obtained by letting successively 9 be one of the parameters... 

Matrix-Based Determination of Parameter Sensitivities of ARR Residuals... 

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. 

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 [14], the author of this chapter briefly showed that incremental bond graphs can also be used to determine parameter sensitivities of the residuals of analytical redundancy relations (ARRs) used in model-based fault detection and isolation. This section elaborates this aspect and gives an illustration. 

For small systems, parameter sensitivities of residuals of ARRs can be manually determined in the following way. First, junctions to which detectors have been attached are identified in the bond graph of the system. The number of structurally independent residuals equals the number of sensors present in the system [13]. Then, virtual detectors are attached to corresponding junctions in the incremental bond graph. Adding variations of flows or efforts, respectively, at these junctions yields variations of residuals of ARRs and thus parameter sensitivities of the residuals. 

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. 

Chapter 4 shows how incremental bond graphs enable a matrix-based determination of parameter sensitivities of transfer functions for direct as well as for inverse linear models. The necessary matrices can be generated from a bond graph and its incremental bond graph by means of existing software. Furthermore, incremental bond graphs also support a parameter sensitivity analysis of ARR residuals. 

Furthermore, studying the effect of component parameter uncertainties on the residuals of ARRs helps in fault isolation. Therefore, Section 4.6 addresses the systematic derivation of parameter sensitivities of residuals of ARRs from an incremental bond graph. 

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. 

An incremental true bond graph approach to a matrix-based determination of parameter sensitivities of transfer functions of linear MIMO models and of residuals of ARRs in symbolic form has been presented. The approach has the following advantages ... 

Adding variations of power variables at junctions of the incremental bond graph immediately leads to parameter sensitivities of residuals of ARRs. 

If the inputs into the system, the measured outputs and the discrete switch states may be considered faultless, gj () becomes a function of the component parameter vector, their rows may be removed from the FSM and the FSM can be considered a structural parameter sensitivity matrix for ARR residuals. 

Faults in a system component can be related to deviations of parameter values from those of the healthy component. For instance, if a hydraulic check valve that autonomously switches on and off is modelled as a non-ideal switch, then a stuck-open fault can be captured by a permanent change in its switch state. Thus, a matrix entry Sij that is non-zero for some system modes means that in these system modes, residual rj is structurally sensitive to faults in the tth component. A FSM thus relates discrepancies in components to changes in residuals. The columns of a FSM indicate the fault signatures of the ARRs of the residuals. Structurally independent ARRs, i.e. ARRs that cannot be algebraically constructed from other ARRs have a unique fault signature. The rows of a FSM are called component fault signatures. An all-mode FSM will be termed hybrid FSM (HFSM). 

The structural fault signature matrix considered so far indicates which component parameters are contained in which ARR. As faults in a component can be related to unwanted changes of parameter values, a structural FSM displays which ARRs are structurally sensitive to which faults. This matrix does not capture that variations in some of the parameters contained in an ARR may have only little effect on the residual of an ARR and may be overshadowed by the affect of other parameter variations in an ARR. As to hybrid models, the sensitivity of an ARR residual with... 

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. 

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. 

Sensitivity analysis of a residual to a parameter uncertainty can be done by deriving the uncertain part a of the ARR according to uncertainty Si as shown by (3.28) and (3.29). The result is a power variable (effort or flow), derived using the nominal value of the parameter. The sensitivity of the ARRs generated from 1 junction and... 

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