Effort source

For illustration, the example of the buck converter is considered once again. As can be seen from the bond in preferred integral causality in Fig. 2.20, there is a causal path from the effort source to the C-element. However, the I-element is in derivative causality. It can be brought into integral causality by inverting the causality at one of the two switches. If the causality at switch Sw m2 is inverted (cf. Fig. 3.2) then there is a direct causal path from the effort source to the I-element that passes through both switches. 

The C-element, however, is not reachable by a direct causal path starting from the effort source. Hence,... 

Its has full rank only if mi = 1 a m2 = 1. However, in this example circuit, the switches commutate conversely. That is, rank(Sj) = 0 for system mode mi = 0 A m2 = 1. From a look at the circuit schematic it is clear that the system is not controllable by the effort source when switch Sw mi is open. However, this cannot be concluded by just evaluating the rank of the structural controllability matrix Sj. 

Inspection of the bond graph in Fig. 2.18 shows that there is a direct causal path from the effort source to the I-element and a causal path from the effort source through the I-element to the capacitor. That is, both storage elements are reachable from the effort source independently of the switch states. The sufficient condition is also satisfied. Both storage elements take derivative causality in the bond graph with preferred derivative causality in Fig. 3.1. Hence, the model of the circuit in Fig. 2.17 with two independent switches is structurally completely state controllable with the one effort source for all four system modes. 

The coupling of the real system model to the faultless model is depicted in the BG fragment of Fig. 4.14 for the case of a residual effort source. The flow f obtained by numerical computation of the real system model may be considered a substitute of a measured flow. For brevity, modulated residual sinks are denoted by rSe or rSf respectively. 

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. 

Constant excitations to a system are represented by an effort or a flow source that provides an output of constant value. In the incremental bond graph these sources are replaced by sources of value zero. If a constant excitation, however, is to be considered uncertain, its source may be replaced in the incremental bond graph by a source modulated by the nominal value. For instance, let Se En represent a constant voltage or constant hydraulic pressure supply. If there is a relative uncertainty 8e = AE/E , then the constant effort source may be replaced in the incremental bond graph by an effort source MSe SsEn modulated by the nominal effort E obtained from the bond graph with nominal parameters. If the internal structure and the parameters of the device are known that provides the excitation and if possible disturbances acting on the device can be modelled, then an incremental bond graph model can be constructed that accounts for the uncertainty of the excitation. 

For sources there is no choice. For an effort source, the output is the effort, for a Sf source, it is the flow. 

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Source nodes All dependent port variables of a source node are independent of its independent port variables. This means that the dependent variables are either constant (linear case with one parameter) or the function of an input (modulated source). This means that a multiport source node can always be split into a set of (modulated) one-port sources. When the dependent port variable is an effort the source is called an effort source (node label Se). When the dependent port variable is a flow the source is called a flow source (node label Sf). A modulated source has node label MSe or MSf. 

A power bond with low relative activity RA at a junction, as defined in Section 2.2, can be conditioned or converted to a modulated source due to the fact that one of the two power variables that are shared by the elements connected by the bond does not contribute significantly to one of the associated constitutive laws. The conditioning is illustrated in Table 2.5. Scenario (i). Case A, is a case where a bond has low RA at a 0-junction. In other words, activity Ai A +i,..., A . Assuming that the low activity is due to relatively low flow in the 0-junction flow summation [24], the low-RA bond is removed from the flow summation by replacing it with a modulated effort source. The modulating signal is the effort out of the junction. This... 

The bond graph model of the nominal system in integral causality is given in Fig. 3.18. The mechanical part of the engine is characterized by the viscous friction fm and inertia Jm- Load part is characterized by friction fs and inertia Js. Reducer part is represented by TF, and the axles stiffness at the input and output of the reducer is represented by C 1/K element. Modulated effort sources d and (U are the disturbing torques caused by the presence of the backlash. Axle velocities are represented on the bond graph model of Fig. 3.18 by two flow sensors Df % and Df ... 

Furthermore, consider an effort-modulated effort source with the constitutive... 

Fig. 4.4 First-order incremental bond graph model of an effort-modulated effort source...

In this example, input-output pairs I t), e and E t), f2, respectively, are collocated. Hence, the left-hand side flow source and the effort detector in Fig. 4.20 can be combined into one source-sensor element SS. The same holds for the right-hand side effort source and the flow detector. The bond graph of the inverse model is... 

The inverse bond graph is obtained from the direct bond graph (Fig. 4.11) by replacing each of the two effort sources representing the voltage source and the external moment by a flow source-effort sensor, SS, as depicted in Fig. 4.24. The source-sensor elements lead to differential causality at the ports of the two I elements accounting for the self-inductance La of the rotor winding and the mechanical inertia Jm of rotor and load. That is, the inverse model has no states. Hence, the denominator of all transfer functions of the inverse model is a constant. 

As indicated in Fig. 5.2b, effort actuation leads to all four dynamic components (Iji. IJ2, C ki and C k2) having integral causality. However, the effect of the effort source is to break the system into two subsystems which, apart from sharing an input, are completely separate. It follows that using effort actuation the system is uncontrollable moreover, as the uncontrollable part is also unstable, the system is unstabilisable. 

Dualise the source thus the effort source becomes a flow source and vice versa... 

Figure 5.3a shows the result of applying this recipe to the system with effort actuation (when dualised, the effort source Se so becomes the flow source Sf so) one component (C ki) is in integral causality and so it is confirmed that this system is not structurally controllable with effort actuation. Conversely, Fig. 5.3b shows the result of applying this recipe to the system with flow actuation no component remains in integral causality and so it is confirmed that this system is structurally controllable with effort actuation. 

The elements representing the element dynamic properties of the system will then be added. These elements are the translational inertance and the rotational inertance of the body, an effort source representing the forces on the center of mass, and an element representing the inertial forces. 

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