Bond graph causal

Section 1.2 gives a literature review of various approaches to a bond graph representation of hybrid system models. In particular, the section discusses the representation of fast switching devices by means of non-ideal switches with fixed mode-independent causality and addresses the equation formulation as a DAE system. 

In order to account for the abstraction of ideal, no power consuming switching in a bond graph with invariant causalities that holds for all system modes, Umarikar extended 0- and 1-junctions by allowing for more than one bond to impose an effort on a 0-junction and more than one bond imposing a flow on a 1-junction with the constraint that only one of the causality imposing bonds is active at a time instant [12, 13], These extensions are called switched power junctions and are not to be confused with controlled junctions to be referred to subsequently. Figure 2.2 illustrates the idea. 

A causality change at a switch port propagates at least locally into the bond graph and affects the causality at the ports of other elements as indicated in Fig. 2.6 for the example of the mechanical stop. 

Back in 1993, Asher proposed to assist an ideal switch by a resistor he called causality resistor that adapts its causality to causality changes at the switch port so that the rest of the bond graph remains causally unaffected [23]. As long as the simulated dynamic behaviour is not significantly affected, the parameter value of a causality resistor can be chosen within reasonable limits but may lead to stiff model equations and thus may give rise to an increase of computational costs. 

The physical meaning of a causality resistor depends on the application area and how it is used in conjunction with an ideal switch. For instance, the bond graph in... 

Fig. 2.7 Bond graph model of an electrical diode using an ideal switch S w and a causality resistor R / 2- a Ideal switch closed, b Ideal switch open...

The causality resistor R / 2 clearly avoids the propagation of causality changes at the port of the ideal switch into the rest of the bond graph and captures the diode s high resistance Roft in reverse mode. The resistor R Ri represents the diode s small ON-resistance Ron-... 

Fig. 2.12 Bond graph representation of a non-ideal switch, a Piecewise linear approximation of the characteristic of a diode, b Bond graph model of a switching device with R Ron in fixed conductance causality...

Fig. 2.15 Bond graph of a clutch with fixed mode independent causalities...

In this book, switching devices such as electrical diodes and transistors, or hydraulic valves are modelled as non-ideal switches represented by a bond graph component model Sw that is composed of a switched MTF and a resistor in fixed conducfance causality. The choice of fixed conductance causality is motivated by the fact that it is the flow through the element that is determined by the discrete switch state. 

In [42], van Dijk has shown that the determinant of det(E ) = det(I — A22), is non-zero for bond graphs with causal paths between resistive ports. That means that the inverse of E exists and that differentiation of the algebraic equation (2.18) is sufficient to transform the DAE system (2.17) into a set ODEs. Accordingly, (2.17) is a DAE system of index 1. 

Fig. 2.18 Bond graph with system mode independent causalities of the electrical circuit in Fig. 2.17...

The bond graph in Fig. 2.18 contains three causal paths between resistors and switches. 

Fig. 2.20 Bond graph in preferred integral causality of the buck converter in Fig. 2.19...

As an example, consider the bond graph of a clutch in Fig. 2.15. There are no causal paths between resistors and no dependent storage elements. Clearly, as long as the clutch is disengaged, the DAE system is of index 0. In the case when the clutch is engaged, the unknown constraint force M between the two plates keeps their inertia elements in integral causality and at the same time ensures that the algebraic constraint... 

Fig. 2.24 Block diagram of a bond graph C storage element in integral causality...

Another approach that also allows to approximate each continuous time element of a bond graph by a DEVS model so that a DEVS simulation can be performed has been reported in [57, 58]. The task is to transform piecewise continuous input and output trajectories of a bond graph element into discrete event trajectories. To translate, e.g. the continuous time model of a C element in integral causality into a discret event model, the input trajectory of the flow f t) between two time instances U and tj is approximated by a linear function f(t) = a t + ao- The output trajectory of e(t) is a second order polynomial e(t) = b2t + bit = aQlC)t. This... 

In this book, a bond graph representation of hybrid system models is chosen as basis for bond graph model-based EDI in Chap. 4. The representation is system mode independent with regard to computational causalities and allows for deriving a set of... 

If the second condition cannot be satisfied so that k = n —q storage elements remain in integral causality in the bond graph with preferred derivative causality (BGD), then k additional sensors are needed to assure complete observability. Their type and their position is to be chosen so that both conditions are satisfied. ... 

Its application to the example circuit in Fig. 2.17 confirms that the circuit with the one voltage sensor is completely observable for all system modes. In the bond graph with preferred integral causality (BGI) (Fig. 2.18), there is a direct causal path from the C-element to the detector and a causal path from the I-element through the C-element to the detector so that condition 1 is satisfied. 

If derivative causality is assigned to the bond graph as preferred causality, (Fig. 3.1), then both storage elements take derivative causality. The resulting causal conflict at the right-hand 0-junction is resolved by turning the effort sensor into a flow sensor. 

In bond graphs of hybrid system models, some storage elements in integral causality may be connected to a detector only via a causal path through a switch model Sw m or even through several switch models. Clearly, the necessary reachability condition is only satisfied when the switch involved in the causal path is closed. That is, a model is not fully state-observable in those system modes in which a switch being part of the only causal path from a storage element to a detector is off, i.e. m = 0 for the switched MTF of that switch model. 

The number q of storage elements that take derivative causality when derivative causality is assigned to a bond graph as preferred causality is termed the bond graph-rank of the state space matrix A and is equal to rank(A) while k equals the number of null modes of A [3]. 

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