# Switched Bond Graphs : Simulation and Analysis

Abstract: In this thesis an approach to modeling and simulation of mode switching systems is investigated. This approach, switched bond graphs, is an extension of the bond graph language in the sense that it allows modeling of mode switching phenomena.The classical bond graph language as well as the switched bond graph language are presented. Different aspects of these tools are discussed, and one aspect specially considered is causality. Computational causality shows in what order the variables in a model should be calculated to get efficient simulation code. Causality can also be used to analyze the model. For classical bond graphs, causality is a fixed property, but for switched bond graphs causality becomes mode varying.In classical bond graphs, algorithms for causality propagation, equation generation, and simulation are well established. Here, corresponding algorithms for switched bond graphs are derived. A causality propagation algorithm is presented, that detects causality problems like algebraic loops, derivative causalities and conflicting causalities. The algorithm checks all modes simultaneously, with a complexity that is proportional to the size of the bond graph, and not to the number of modes. A simulation algorithm is presented, that derives the information necessary for simulating a mode, when the mode is reached in a simulation run. This algorithm circumvents the combinatorial explosion in the number of modes. An algorithm for deriving the complete mathematical description from a bond graph is also given.The algorithms are analyzed, and their claimed functionality is proved. The proofs are based on properties of classical bond graphs. Then it is shown that a class of classical bond graphs can be expressed in state space form, by a variable transformation. For the same class of classical bond graphs, results are given, showing when an algebraic loop occurs, and what variables are included.These results are used to verify a part of the simulation algorithm. It is shown that correct initial values of the continuous state variables are achieved, when a new mode is entered during simulation of a switched bond graph. This problem is non-trivial since the mode varying causality may give rise to discontinuities in the state variables at a change of modes. It is also shown that the presented causality propagation algorithm for switched bond graphs detects all possible causal conflicts and derivative causalities, and the largest possible algebraic loops, in any mode.The initialization of modes is also analyzed in a singular perturbation theory framework, and this analysis verifies the correctness of the mode initialization algorithm.Finally, the simulation algorithm is applied to two examples.

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