Numerical simulation of continuous systems with structural dynamics

In this paper, "continuous systems with structural dynamics" shall be understood asdynamical systems consisting of components with continuous and/or discrete behaviour. (This notation should not be confused with the term "structural dynamics" in the context of Finite Element Simulation). Continuous systems with structural dynamics - or so called "hybrid systems" - can often be investigated only by a so-called "hybrid simulation" which means a simultaneous simulation of continuous-time dynamics (modelled by differential equations or differential-algebraic equations (DAE) and discrete-event dynamics (modelled e.g. by Boolean equations, finite state machines, or statecharts). To this end, an algorithm for numerical simulation of hybrid systems must be able to both solve a DAE system within a "continuous" time progression as well as to deal with event-driven phenomena. In the paper, the point of view is emphasized that the structure of a continuous system is closely combined to the structure of the DAE system which describes the continuous system's dynamical behaviour. In this context, discrete-time events are considered as phenomena which may cause a change of the DAE system's structure. Furthermore, a distinction between systems with variable structure and models with variable structure is explained. The main part of the paper deals in detail with a simulation algorithm suitable for hybrid systems. This algorithm consists of a "continuous phase" (for numerical integration of the DAE system) and a "discrete phase" (for interpreting the event, establishing the new valid DAE system, calculating the new initial values). Some simulation results dealing with selected models and using the multi-physics language Modelica will complete the paper.