Hammerstein system identification by non-parametric instrumental variables

Abstract

A mixed, parametric–non-parametric routine for Hammerstein system identification is presented. Parameters of a non-linear characteristic and of ARMA linear dynamical part of Hammerstein system are estimated by least squares and instrumental variables assuming poor a priori knowledge about the random input and random noise. Both subsystems are identified separately, thanks to the fact that the unmeasurable interaction inputs and suitable instrumental variables are estimated in a preliminary step by the use of a non-parametric regression function estimation method. A wide class of non-linear characteristics including functions which are not linear in the parameters is admitted. It is shown that the resulting estimates of system parameters are consistent for both white and coloured noise. The problem of generating optimal instruments is discussed and proper non-parametric method of computing the best instrumental variables is proposed. The analytical findings are validated using numerical simulation results.

Keywords:

• Hammerstein system
• parameter estimation
• instrumental variables
• non-parametric regression
• convergence analysis

Notes

Note

1. From now on the Euclidean norm will be denoted by ‖ · ‖₂ to avoid ambiguity.