Bond graph sensitivity

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. 

Samantaray, A. K., Ghoshal, S. K. (2007). Sensitivity bond graph approach to multiple fault isolation through parameter estimation. Proceedings of the Institution ofMechanical Engineers Part I Journal of Systems and Control Engineering, 221(4), 577-587. 

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. 

Gawthrop, P. J. (2000). Sensitivity bond graphs. Journal of the Franklin Institute, 337,907-922. 

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. 

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. 

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. 

Bomtzky, W., Granda, J. J. (2002). Bond graph based frequency domain sensitivity analysis of multidisciplinary systems. Proc Instn Mech Engrs, Part I, Journal of Systems and Control Engineering, 216(1), 85-99. 

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]. 

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. 

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. 

In the following, first, sensitivity pseudo bond graphs are briefiy reviewed and are then used to obtain residual sensitivity functions needed for the previously presented least squares ARR residuals minimisation. The simple hybrid network in Fig.4.1 is used again for illustration of the approach. 

Parameter Sensitivity Models of Bond Graph Elements... 

The sensitivity component bond graph model in Fig. 6.16 reduces to a 1-port resistor if 0 Or. 

Similarly, a sensitivity component bond graph model is obtained for a nonlinear 1-port C storage element in derivative causality given by the constitutive relation... 

Fig. 6.16 Parameter sensitivity component bond graph model MR of a nonlinear 1-port R element...

Figure 6.17 displays a sensitivity component bond graph model MC according to (6.29). 

Like in the previously considered case of a resistor, the modulated flow sink disappears in the sensitivity bond graph model of a C storage element if 0 does not equal the element parameter... 

In the same manner, sensitivity component models can be obtained for the other bond graph elements. As junctions do not depend on parameters they remain junctions in a sensitivity pseudo bond graph. Sources that provide a constant become sources of value zero. Sensitivity component models of other elements differ from their element only by additional sinks. As a result, a sensitivity pseudo bond graph is of the same structure as the behavioural system bond graph. Moreover, causalities of the latter one are retained. 

Deducing Residual Sensitivity Functions from a Sensitivity Pseudo Bond Graph... 

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. 

Gawthrop, P. J. (2000). Sensitivity bond graphs. Journal of the Franklin Institute, 337,907-922. Gawthrop, P.J., Ronco, E. (1999). A sensitivity bond graph approach to estimation and control of mechatronic systems. Centre for systems and control. University of Glasgow, Faculty of Engineering, CSC-99018. 

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. 

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