Switched residual sink

Switching Off and On Degrees of Freedom by Means of Switched Residual Sinks... 

For a system mode, in which such switched residual sinks provide a non-zero output k, the DAEs for a switched LTI system are of the form... 

As a result, if the dynamic behaviour of a system can be described by a switched LTI system, the DAE system derived from the bond graph is of index <1 as long as no structural changes occur. If a structural change modelled by a switched residual sink happens, the DAE index jumps to two. 

If, for some time intervals, no residual sinks are switched on via an MTF, then the mathematical model takes the form... 

First, it is assumed that all storage elements are in integral causality for all system modes. That is, no residual sinks are switched on at discrete events to keep some storage elements in integral causality that otherwise would be become dependent and would get derivative causality accordingly. Causal paths between resistive ports are allowed. As the switch model contains a resistor in conductance causality, there may also be causal paths between a resistor and a switch or between switches. If a switch in one of these causal paths is in OFF mode, the switch and the causal path can be disregarded. Causal paths between resistors mean that their outputs are determined by a set of algebraic relations. Let a denote the vector of the outputs of resistors and of switches, then the DAE of a switched LTI system is of the form... 

For some systems with structural changes such as a clutch, storage elements may temporarily become dependent for the duration of a system mode. In such a case, a residual sink may be switched on that delivers a power variable so that the conjugate power variable vanishes and storage elements can keep integral causality. As their state variables jump to a new joint value, numerical integration has to be re-initialised at such a discrete event. 

If the dynamic behaviour of a system can be described by a switched LTI system, a linear implicit DAE system can be derived from the bond graph. The entries of its matrices depend on the discrete switch states. As long as no structural changes occur, i.e. no residual sinks are switched on, the DAE system is of index <1. For system modes in which residuals sinks are switched on, the DAE system is of index 2. There are solvers available for its direct numerical computation that are based on the BDF-method. An alternative may be to perform a DEVS simulation that uses quantised based integration. 

Remark 4.4 Derivation of equations from the bond graphs coupled by residual sinks results in a DAE system as there are no time derivatives of the outputs of the coupling residual sinks. Let some elements of the switched models have non-linear... 

For illustration, Fig. 4.16 displays the coupling of a real system model and a faultless model by residual sinks for the switched circuit in Fig. 4.1. 

Finally, the coupling of two bond graph models in preferred integral causality by means of residual sinks can be complemented by some analytical formulation in case the behaviour of the real system can be sufficiently accurately modelled by means of a switched LTI system and if the bond graph is in preferred integral causality for all system modes. 

The last two cases require an extension of the load model. The residual sinks introduced in Chap. 2, allow one to keep preferred integral causality when storage elements become dependent. In this application, a switched residual effort sinkrSe A. is used that imposes an effort X onto the two inductor elements I La and I Lc so that their output currents become equal. Figure8.24 shows an extended BG that accounts for a mode dependent number of states. 

A number of research works on Z-source inverters has been reported in the literature (see, for instance, [42-44]). A bond graph model of a mono-phase Z-source inverter with a series RL load has been presented in [45]. The approach uses switched power junctions (SPJs) [20] and residual sinks [46]. 

Nacusse, M. A., Junco, S. J. (2011). Simplifying switched bond graphs using residual sinks to enforce causality application to modeling the zsource inverter, mecanica computacional, industriai appiications (B). 2011 XXX(33) 2533-2548. Retrieved online at http //www.cimec. org.ar/ojs/index.php/mc/article/view/3931. 

Equations can be formulated so that parameters of auxiliary elements can be set to zero. Also, the ON-resistance of switches can be set to zero turning them into ideal switches so that small time constants and thus a set of stiff model equations can be avoided. In the case residual sinks are used, which is similar to the use of Lagrange multipliers, the resulting mathematical model is a DAE system of index 2. 

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. 

When the residual effort sink is activated at time instant to, i.e. when the upper switch opens while the lower switch is permanently off, the two load currents iia and iic jump to a common value is so that ia = 0. This jump at time instance to means that the common value is is to be determined and that numerical integration of the model equations must be re-initialised at to. When the upper switch abruptly closes at time instances tc while the lower switch is still off, then both currents and iLa start from their common value. 

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