Boosting the Model Discovery of Hybrid Dynamical Systems in an Informed Sparse Regression Approach

Abstract We present an efficient data-driven sparse identification of dynamical systems. The work aims at reconstructing the different sets of governing equations and identifying discontinuity surfaces in hybrid systems when the number of discontinuities is known a priori. In a two-stage approach, we first locate the switches between separate vector fields. Then, the dynamics among the manifolds are regressed, in this case by making use of the existing algorithm of Brunton et al. (2016, “Discovering Governing Equations From Data by Sparse Identification of Nonlinear Dynamical Systems,” Proc. Natl. Acad. Sci., 113(15), pp. 3932–3937). The reconstruction of the discontinuity surfaces comes as the outcome of a statistical analysis implemented via symbolic regression with small clusters (microclusters) and a rigid library of models. These allow to classify all the feasible discontinuities that are clustered and to reduce them into the actual discontinuity surfaces. The performances of the sparse regression hybrid model discovery are tested on two numerical examples, namely, a canonical spring-mass hopper and a free/impact electromagnetic energy harvester (FIEH), engineering archetypes characterized by the presence of a single and double discontinuity, respectively. Results show that a supervised approach, i.e., where the number of discontinuities is pre-assigned, is computationally efficient and it determines accurately both discontinuities and set of governing equations. A large improvement in the time of computation is found with the maximum achievable reliability. Informed regression-based identification offers the prospect to outperform existing data-driven identification approaches for hybrid systems at the expense of instructing the algorithm for expected discontinuities.