Scaling of the sampling period in nonlinear system identification

Abstract

The paper presents a scaling algorithm for system identification, based on a nonlinear black box differential equation model. The model is discretized by an Euler forward numerical integration scheme. A scale factor is applied to the explicitly appearing sampling period, when iterating the discrete time state space model and the corresponding gradient recursion. The result is an exponential scaling of the state components of the model, and a scaling of the estimated parameter vector. The original parameter vector can be explicitly calculated from the scaled parameter vector using a diagonal matrix that is a function only of the scale factor. A new analysis of the effect of scaling on the Hessian, shows how the same diagonal matrix affects its eigenvalue distribution. A simulation study illustrates the beneficial effects on e.g. the condition number that can be obtained with the algorithm.