Incremental bond graph

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. 

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. 

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. 

An incremental bond graph can be constructed in a systematic manner from the original bond graph of a switched LTI system by replacing an element that is due to parameter variations by its incremental element model. Equations for variations of power variables can be automatically derived in the same way as they are derived from an initial bond graph with nominal parameters. 

Switched LTI systems are just LTI systems for the time intervals betweens between discrete mode changes. Therefore, first, the incremental bond graph approach is recalled for LTI systems. In a second step, an incremental model for switches is... 

The underlying idea of incremental bond graphs is that if a parameter 0 of a component model varies, then both power variables at its port are perturbed due to its interaction with the ports of other elements in the model. That is, an effort (f) in a bond graph with nominal parameters becomes e t) = en t) + Ae t). The same holds for the conjugate power variable /(r). In incremental bond graphs, bonds carry the increments Aeif), Af(l) of power variables. In other words, they represent energy flows carrying the amount of power Ae t) Af(t). 

An incremental bond graph of an LTI system is constructed from an initial bond graph with nominal parameters by replacing those bond graph elements by their incremental model for which a parameter variation has taken place. Clearly, sources that do not depend on a parameter become null sources. Furthermore, the incremental model of a l(0)-junction remains a l(0)-junction. As to the incremental models of storage elements, resistors, transformers and gyrators, it turns out that they differ from the respective element just by modulated sinks added to junctions. The sinks... 

In order to see how an incremental bond graph model for a bond graph element is obtained, a linear 1-port C-element with the nominal capacitance Cn is considered. In the following, an index n indicates a parameter or a variable of the non-faulty bond graph model with nominal parameters. In the case of a time constant parameter variation AC the linear constitutive equation of a 1-port C element in derivative causality takes the form... 

Figure 5.1 shows a bond graph representation of (5.2). In essence, the incremental model of a C element is again a C element. A modulated sinkMSe Sc fc contributes to its output Afc. 

The power variable fc controlling the modulated source is an output variable of the original bond graph model. If fc has been obtained by measurements of the real system, then the contribution to the output of the incremental bond graph model of the C element may contain sensor noise. In any case, the outputs of the incremental bond graph of a bond graph element indicate a parameter variation. 

Fig. 5.1 Incremental bond graph model of a linear 1-port C element in preferred derivative causality...

In the same manner, incremental models for the other bond graph elements may be obtained. As an example. Fig. 5.3 depicts the incremental bond graph model of a transformer TF ... 

Incremental Models of Nonlinear Bond Graph Elements... 

The incremental bond graph approach is not limited to linear 1 -port elements with one parameter [5]. The constitutive equation of the incremental model of a bond graph element can be easily obtained by taking the total differential of the constitutive equation of the bond graph element. 

Equation 5.6 may be represented by an incremental bond graph model with nonUnear... 

Fig. 5.6 Extended incremental bond graph model of a non-ideal switch accounting for a non-zero flow in OFF-mode...

If the flow close to zero in OFF-mode is not neglected, then the switch in OFF-mode can be considered a resistor with a very high OFF resistance Roir. For this resistor, an incremental model can be developed in the same manner as for the resistor R f on In the resulting model, the resistor R Roff can be neglected as the nominal value of 1/Roff is very small. Figure5.6 depicts an extended incremental bond graph model of a non-ideal switch that accounts for a non-zero flow in OFF-mode. 

Furthermore, let 0 denote the vector of aU component parameters. The state space model derived from the incremental bond graph then is of the form... 

The matrices A and C are obtained deriving equations from the original bond graph with nominal parameters. Matrix B can be automatically set up from the incremental bond graph. The vector w denotes the outputs of the modulated sinks in the incremental bond graph that represent parameter variations. These outputs are of the form... 

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. 

As an example, the circuit with one switch in Fig. 4.1, is considered. To keep the illustration of the procedure short and simple it is assumed that only one parameter is uncertain. Accordingly, the incremental bond graph is obtained by replacing the element by its incremental model and by replacing the constant voltage source Se Vi by an effort source of value zero and by replacing detectors by dual virtual detectors for the variations of ARR residuals. 

Figure 5.8 displays the corresponding incremental bond graph. Again, the purpose of the auxiliary storage element C Q is just to resolve the causal conflict at junction O2. In the process of equation formulation, the capacitance Ca is set to zero. Summing variations of power variables at junctions li, O2, and is yields... 

Fig. 5.8 Incremental bond graph of the switched circuit in Fig. 4.1 in the case of an uncertain parameter R ...

Figure 5.9 displays the corresponding incremental bond graph. Summation of flow variations at junction Oe gives... 

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