Linear approximations of nonlinear FIR systems for separable input processes

  • Authors:

  • Martin Enqvist;Lennart Ljung

  • Affiliations:

  • Division of Automatic Control, Department of Electrical Engineering, Linköping University, SE-58183 Linköping, Sweden;Division of Automatic Control, Department of Electrical Engineering, Linköping University, SE-58183 Linköping, Sweden

  • Venue:
  • Automatica (Journal of IFAC)
  • Year:
  • 2005

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Abstract

Nonlinear systems can be approximated by linear time-invariant (LTI) models in many ways. Here, LTI models that are optimal approximations in the mean-square error sense are analyzed. A necessary and sufficient condition on the input signal for the optimal LTI approximation of an arbitrary nonlinear finite impulse response (NFIR) system to be a linear finite impulse response (FIR) model is presented. This condition says that the input should be separable of a certain order, i.e., that certain conditional expectations should be linear. For the special case of Gaussian input signals, this condition is closely related to a generalized version of Bussgang's classic theorem about static nonlinearities. It is shown that this generalized theorem can be used for structure identification and for the identification of generalized Wiener-Hammerstein systems.