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The interacting multiple model filter has long been the method of choice for performing target tracking using multiple motion models. The filter finds a suboptimal solution to a problem which has the implicit assumption that immediate model shifts have the highest probability. When the sampling rate of the underlying continuous process is high compared to the target dynamics, this is not a reasonable assumption. Instead, changes in dynamics persist for some time. In this paper we propose an alternative switching model, which forces the dynamic models to persist for at least a model-specific time. The model is semi-Markov in nature, with a sojourn time probability mass function which is zero for a model-specific number of time steps, and then follows a geometrical distribution. Through this assumption a less complex problem in terms of model hypotheses arises, and to that problem we derive a state estimation algorithm that is close to optimal when the model assumptions are valid. Three other semi-Markov-based multiple-model filters are discussed and compared to in a qualitative sense. We also derive a new aircraft motion model for start and termination of turns. Finally, the proposed filter is evaluated on a benchmark scenario for tracking, and the results show a performance increase compared to the interacting multiple model (IMM) filter for the trajectories considered.