Analysis of Causal Bond Graph Models

Abstract

So far, we have seen that causalities can be assigned to a bond graph by differ-ent methods (SCAP, relaxed causalities, Lagrange causalities) and that different forms of mathematical time domain models can be derived from a bond graph (state space form, descriptor form, Lagrange equations). Since mathematical models de-rived from bond graphs frequently take the form of a DAE system, its index and approaches to a symbolic and numerical solution have been considered. However, not only simulation of the dynamic behaviour of a multidisciplinary system is of concern, but other tasks also, e.g., the determination of a steady-state needed for the linearisation of the model equations, the establishment of transfer functions, the determination of pole-zero loci. Moreover, with regard to the design of a controller, properties, e.g., structural controllability and observability, are of interest. Of course, once a time domain model is available, the information needed can be derived from the linearised model equations. In this chapter, we will see that not only time domain models but other infor-mation relevant for control system design can be derived directly from a causally completed bond graph. That is, bond graphs can be viewed as a core model repre-sentation from which information for different purposes can be derived in suitable form. According to this view, Gawthrop and Smith developed a set of model trans-formation tools collected in a toolbox MTT (Model Transformation Tools) [35]. Depending on the actual task, these tools enable one to automatically transform one model representation into another where an acausal bond graph is the core rep-resentation. Aspects of this approach to automated modelling will be considered separately in Section 11.6.6. In the next section, it is shown how a bond graph can be used to set up equations for the determination of the steady-state of a dynamic system. 223 224 6 Analysis of Causal Bond Graph Models 6.1 Equations Determining the Steady-state of a Dynamic System In bond graph terms, the steady-state of a dynamic system is characterised by the fact that flows into C energy stores and efforts into I elements are equal to zero. This can be expressed in a bond graph in two different ways. First, a C energy store can be replaced by a flow sink that imposes a flow equal to zero. The value of the effort into the sink is the steady-state to be determined. Alternatively, a C energy store can be replaced by an effort sink imposing an unknown constant effort such that the flow into the sink is equal to zero. Reverse statements hold for an I element dual to a C energy store. The first option corresponds to the approach adopted in circuit analysis. For the determination of the steady-state, capacitors in network are removed and inductances are replaced by short circuits. In bond graphs, the substitution of energy stores entails a reassignment of causalities. As a consequence, it can happen that a flow sink replacing a C energy store attached to a 0-junction does not determine the common effort at that junction anymore. In the same way, an effort sink replacing an I element attached to a 1-junction does not determine the common flow at that junction. Consequently, the causality at some resistive port or at an internal bond must be changed. Thus, a causal path between resistive ports or a causal loop may emerge. On the contrary, if C energy stores are replaced by constant effort sinks and I elements by constant flow sinks, then causalities are retained. In both cases, algebraic equations determining the steady-state can be derived from the modified bond graph. The resulting sets of equations are equivalent, but different in form due to different causalities in the bond graph.