Some Results on Linear Models of Nonlinear Systems

Sammanfattning: Lineartime-invariant approximations of nonlinear systems are used in manyapplications. Such approximations can be obtained in many ways. Forexample, using system identification and the prediction-errormethod, it is always possible to estimate a linear model withoutconsidering the fact that the input and output measurements ingeneral come from a nonlinear system. The main objective of thisthesis is to explain some properties of such estimated models.More specifically, linear time-invariant models that are optimalapproximations in the mean-square error sense are studied.Although this is a classic field of research, relatively fewresults exist about the properties of such models when they arebased on signals from nonlinear systems. In this thesis, someinteresting, but in applications usually undesirable, properties oflinear approximations of nonlinear systems are pointed out. It isshown that the linear model can be very sensitive to smallnonlinearities. Hence, the linear approximation of an almostlinear system can be useless for some applications, such as robustcontrol design. In order to improve the models, conditions are given on the inputsignal implying various useful properties of the linearapproximations. It is shown, for instance, that minimum phasefiltered white noise in many senses is a good choice of inputsignal. Furthermore, some special properties of Gaussian signalsare discussed. These signals turn out to be especially useful forapproximations of generalized Hammerstein or Wiener systems. Usinga Gaussian input, it is possible to estimate the denominatorpolynomial of the linear part of such a system without compensatingfor the nonlinearities. In addition, some theoreticalresults about almost linear systems and about separable inputprocesses are presented. Linear models, both with and without anoise description, are studied.