A randomized two-stage iterative method for switched nonlinear systems identification

This paper addresses the identification of discrete time switched nonlinear systems, which are collections of discrete time nonlinear continuous systems (modes) indexed by a finite-valued variable defining the current mode. In particular, we consider the class of Switched Nonlinear AutoRegressive eXogenous (Switched NARX, or SNARX) models, where the continuous dynamics are represented by NARX models. Given a set of input–output data, the identification of a SNARX model for the underlying system involves the simultaneous identification of the mode sequence and of the NARX model associated to each mode, configuring a mixed integer non-convex optimization problem, hardly solvable in practice due to the large combinatorial complexity. In this paper, we propose a black-box iterative identification method, where each iteration is characterized by two stages. In the first stage the identification problem is addressed assuming that mode switchings can occur only at predefined time instants, while in the second one the candidate mode switching locations are refined. This strategy allows to significantly reduce the combinatorial complexity of the problem, thus allowing an efficient solution of the optimization problem. The combinatorial optimization is addressed using a randomized method, whereby the sample-mode map and the SNARX model structure are characterized by a probability distribution, which is progressively tuned via a sample-and-evaluate strategy, until convergence to a limit distribution concentrated on the best SNARX model of the system generating the observed data.