The Identification of Non-linear Systems with Statistically Equivalent Polynomial Systems

Abstract

Volterra (or Wiener) series models of non-linear systems have been studied by many authors, e.g. Rugh[1], Tomlinson[2]. By means of the Multi-dimensional Frequency Response Function (MFRF) based on the Volterra (or Wiener) kernel the non-linear dynamical systems can be analysed in the frequency domain. However, there are some crucial problems concerning the estimation and application of MFRFs[3]. The first is the expense of estimating MFRFs by direct methods, which impels us to study the parameter estimation methods. Billings[4] has developed the estimation method of MFRF by means of the NARMAX model. The polynomial approach for estimating MFRF is discussed in detail in the paper. From the point of view of the estimation of MFRF, the NARMAX model looks more convenient because the output relates directly to the input. However, from the point of view of the discussion on dynamics behaviour, there are more advantages with the parameter identification of the polynomial non-linear model. This is because it is used together with the equation of motion of the dynamic system. Natke and Zamirowski[5] discussed the method of structure identification for the class of polynomials within mechanical systems, which laid a foundation for the parameter estimation method of MFRF. The second crucial problem is that the difficulties arise from using MFRF to describe the non-linear properties of the system because the multi-frequency is without a physical meaning. In the second part of this paper an attempt is made to decrease these difficulties. While the standardized formula of the parameter estimation is given with the response and the excitation, emphasis is given to the study of the statistical equivalent 3rd order polynomial system, the spectral structure of the response expressed with MFRF is analysed in detail, and the Extended Transfer Functions (ETF) are defined, which are only functions of one-dimension frequency. In addition, the effectiveness of the statistical equivalence is analysed. The advantages of the method used here are that the MFRF of the polynomial non-linear system has the theoretical analytical expression, and by studying a non-linear system particular properties with extended transform functions avoid the difficulty of graphing the MFRF.