Event Detection and Localization

Event detection is executed at each integration time step. Only threshold functions of signal ports that may cause a state change are evaluated. Comparison is based on the sign (negative, 0, positive) of the signal port function value. The current value is stored and if a change of sign is detected, this is flagged. Note that a small tolerance around 0 is applied to handle numerical imprecision. Event detection may fail when the comparison function has an even number of roots in the one time interval of numerical integration. However, because the comparison function applies signals directly generated by the dynamic model behavior, it is likely that such an interval is too large for simulating the continuous dynamics to begin with.

When an event is detected, it is localized by a bisectional search. This is a robust technique that finds the first time of occurrence of an event within a prespecified margin [Cellier1979]. Next, a corresponding time step is made, $x = timeStep(\alpha, x)$ in Algorithm 1. After this, enabled state changes of all junctions are executed and given the new hybrid bond graph structure, the model variables are recomputed. If algebraic constraints between state variables or between state variables and sources become active, this may cause a discontinuous change in state variable values. The new model variable values may cause further events to occur and corresponding junction state changes are executed. The state vector is restored to its value that triggered the very first junction state change and based on this state the new hybrid bond graph model variable values are re-calculated. This iteration proceeds until no further junction state changes occur, $[\alpha^+, x^+] = iterate(\alpha, x)$ in Algorithm 1. If a change between the last mode of continuous behavior, $\alpha$, to a new mode, $\alpha^+$, has occurred, the state vector is updated. In case of modeled time scale abstraction, this may again cause further events to occur, and the event iteration starts anew. Once event iteration has converged and updating the state vector causes no further junction state changes, continuous simulation resumes. Details of the hybrid system simulation algorithm are presented elsewhere [Mosterman1997,Mosterman & Biswas1997b].