Shooting methods for identification of nonlinear state-space grey-box models

Identifying state-space models that are nonlinear in the parameters is still a challenge due to the non-convex nature of the objective function and possible regions of the parameter space where the model is non-contractive, even if the model structure is known in advance from physical knowledge. The optimization problem for estimating the model parameters can be formulated in different ways, including single shooting (unconstrained), hybrid shooting (partially constrained) or multiple shooting (fully constrained) formulations, and if the states can be measured or inferred, the estimation can also be formulated as an N-step-ahead prediction problem. In this paper, we compare these methods with respect to running time and robustness against initial parameters being in a non-contractive region and a contractive region, applied to an electromechanical positioning system. We show that for this system with states that can be partially measured and partially inferred, a two-step approach that involves the prediction error formulation and single shooting, is the most robust with respect to non-contractive initialization. The methods described in this paper are released as an open-source MATLAB toolbox that uses algorithmic differentiation and parallelization to speed up the computations.