A bilevel programming framework for piecewise affine system identification

In this paper we propose a bilevel programming formulation of the piecewise affine regression problem, where the upper level fixes the partition of the regressor domain and classifies the data points, and the lower level computes parameter estimates of the affine models in each region of the partition. The proposed formulation accommodates a general class of regression problems, where parameter estimates in each region of the partition are based on a prediction error criterion, while the overall piecewise affine model is selected according to a possibly different criterion. Due to the use of binary variables for the classification task, the proposed approach is typically viable only for small data sets and number of submodels. Still, it can be used to formulate and solve several problems of practical interest, where it is important to carry out simultaneously the three tasks of data classification, parameter estimation and estimation of the partition of the regressor domain. This is demonstrated through an application of the proposed approach to the pick-and-place machine data set.