Nonlinear system identification via direct weight optimization

Authors:
Jacob Roll;Alexander Nazin;Lennart Ljung
Affiliations:
Division of Automatic Control, Linköping University, SE-581 83 Linköping, Sweden;Institute of Control Sciences, Profsoyuznaya str., 65, 117997 Moscow, Russia;Division of Automatic Control, Linköping University, SE-581 83 Linköping, Sweden
Venue:
Automatica (Journal of IFAC)
Year:
2005

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Abstract

A general framework for estimating nonlinear functions and systems is described and analyzed in this paper. Identification of a system is seen as estimation of a predictor function. The considered predictor function estimate at a particular point is defined to be affine in the observed outputs and the estimate is defined by the weights in this expression. For each given point, the maximal mean-square error (or an upper bound) of the function estimate over a class of possible true functions is minimized with respect to the weights, which is a convex optimization problem. This gives different types of algorithms depending on the chosen function class. It is shown how the classical linear least squares is obtained as a special case and how unknown-but-bounded disturbances can be handled. Most of the paper deals with the method applied to locally smooth predictor functions. It is shown how this leads to local estimators with a finite bandwidth, meaning that only observations in a neighborhood of the target point will be used in the estimate. The size of this neighborhood (the bandwidth) is automatically computed and reflects the noise level in the data and the smoothness priors. The approach is applied to a number of dynamical systems to illustrate its potential.