On the Smoothness of Nonlinear System Identification

New light is shed onto optimization problems resulting from prediction error parameter estimation of linear and nonlinear systems. It is shown that the smoothness" of the objective function depends both on the simulation length and on the decay rate of the prediction model. More precisely, for regions of the parameter space where the model is not contractive, the Lipschitz constant and $\beta$-smoothness of the objective function might blow up exponentially with the simulation length, making it hard to numerically find minima within those regions or, even, to escape from them. In addition to providing theoretical understanding of this problem, this paper also proposes the use of multiple shooting as a viable solution. The proposed method minimizes the error between a prediction model and observed values. Rather than running the prediction model over the entire dataset, as in the original prediction error formulation, multiple shooting splits the data into smaller subsets and runs the prediction model over each subdivision, making the simulation length a design parameter and making it possible to solve problems that would be infeasible using a standard approach. The equivalence with the original problem is obtained by including constraints in the optimization. The method is illustrated for the parameter estimation of nonlinear systems with chaotic or unstable behavior, as well as on neural network parameter estimation.