Identification of piecewise affine systems using a cluster refinement technique

The identification of piecewise affine (PWA) systems is a challenging mixed integer optimization problem that involves both the estimation of the dynamics associated to different modes of operation, and the partition of the state space in regions associated to said modes, the transition from one region to another corresponding to a mode switching. The challenges are mainly associated with the sample-mode assignment task, because the combinatorial complexity increases with the size of the dataset. Furthermore, some samples are consistent with more than one mode, making their classification ‘ambiguous’. The identification problem is here addressed with a two-stage iterative method, alternating between an identification phase carried out over given clusters of data associated to regions in the state space (such that each cluster is assigned to a single mode), and a refinement phase, whereby the region borders are adjusted (by reassigning samples to other clusters) to improve the model quality. Operating on data clusters (as opposed to individual samples) significantly reduces the complexity of the combinatorial mode assignment problem, and naturally avoids region outliers (isolated samples surrounded by samples assigned to a different mode). However, this approach works properly only if accompanied by a cluster refinement procedure, responsible for reshaping the mode regions and reassigning stray samples to the correct modes. The combination of these two stages is ultimately successful in determining correctly both the local models and the associated state space regions, as shown here with reference to several benchmark examples.