Some Results on Linear Models of Nonlinear Systems

• Linear time-invariant approximations of nonlinear systems are used in many a pplications. Such approximations can be obtained in many ways. For example,using system identification and the prediction-error method, it is always possible to estimate a linear model without considering the fact that the input and output measurements in general come from a nonlinear system. The main objective of this thesis is to explain some properties of such estimated models.More specifically, linear time-invariant models that are optimal approximations in the mean-square error sense are studied. Although this is a classic field of research, relatively few results exist about the properties of such models when they are based on signals from nonlinear systems. In this thesis, some interesting, but in applications usually undesirable, properties of linear approximations of nonlinear systems are pointed out. It is shown that the linear model can be very sensitive to small nonlinearities. Hence, the linear approximation of an almost linear system can be useless for some applications, such as robust control design.In order to improve the models, conditions are given on the input signal implying various useful properties of the linear approximations. It is shown, for instance, that minimum phase filtered white noise in many senses is a good choice of input signal. Furthermore, some special properties of Gaussian signals are discussed. These signals turn out to be especially useful for approximations of generalized Hammerstein or Wiener systems. Using a Gaussian input, it is possible to estimate the denominator polynomial of the linear part of such a system without compensating for the nonlinearities. In addition, some theoretical results about almost linear systems and about separable input processes are presented. Linear models, both with and without a noise description, are studied.

Nyckelord

System identification

Linear models

Nonlinear systems

TECHNOLOGY

TEKNIKVETENSKAP

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